NCERT Exemplar for Class 11 Maths - Permutations and Combinations - Free PDF Download
Free PDF download of NCERT Exemplar for Class 11 Math Chapter 7 - Permutations and Combinations solved by expert Math teachers on Vedantu as per NCERT (CBSE) Book guidelines. All Chapter 7 - Permutations and Combinations exercise questions with solutions to help you to revise the complete syllabus and score more marks in your examinations.
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Access NCERT Exemplar Solutions for Class 11 Mathematics Chapter 7 - Permutations and Combinations (Examples, Easy Methods and Step by Step Solutions)
Examples:
Short Answer Type Questions:
Example 1: In a class, there are
Ans : Given: Total number of boys
Total number of girls
Number of boys to be selected
Number of girls to be selected
Use the formula
To select
To select
Hence, the total number of ways for the teacher to make the selection
Example 2: (i) How many numbers are there between
Ans: Given: The numbers between
fix the digit
To find the numbers between
The hundreds of places can be filled with any digit other than
Hence, the total numbers of numbers
(ii) How many numbers are there between
Ans: Given: The numbers between
form all the three digit numbers that can have
Number of numbers where at least one of digits is
Total number of three digit numbers
Now, in the numbers having no digit as
Hence, the total number of numbers
Example 3: In how many ways can this diagram be coloured subject to the following two conditions?
Each of the smaller triangles is to be painted with one of three colours: red, blue or green And No two adjacent regions have the same colour.

Ans : Given: The diagram: -

Total number of different colours
Fill the middle triangle with any of the three colours. Now, fill the three other triangles with the other two colours (may be same or different).
First let us fill the smaller triangle in the middle. There are
So there are
Hence, the total number of ways to colour all the smaller triangles
Example 4: In how many ways can
(i) two particular children of them are always together.
Ans: Given: Total number of children
assume the two particular children as one unit and arrange them among themselves. Now, arrange them will the remaining three children.
Since two particular children are always together so they can be arranged among themselves in
Now, they can be considered as one unit so the other three children and this one unit can be considered as
Hence, the total number of arrangements
(ii) two particular children of them are never together.
Ans: Given: Total number of children
find the number of possible arrangements if no conditions are applied and subtract the arrangements obtained in (i) from this.
If there are no conditions then the
Hence, the total number of arrangements
Example 5: If all permutations of the letters of the word AGAIN are arranged in the order as in a dictionary. What is the
Ans : Given: The word AGAIN.
Arrange the letters in alphabetical order and find the number of words starting with the letters A, G and I. Check the sum and accordingly find the
The alphabetical order of letters in the word AGAIN is A (repeated twice), G, I, N.
If A is chosen as the first letter of the word then the remaining four letters can be arranged in
If G is fixed as the first letter then the remaining four letters can be arranged in
Similarly, if I is the first letter then also the effective different number of words formed
So a total of
Hence, the
Example 6: In how many ways
Ans : Given: Number of mathematics books
Number of history books
Number of chemistry books
Number of biology books
Arrange all the different subject books individually among themselves. Now, consider the similar subject books as a single unit and arrange them.
Now, the group of similar subject books can be assumed to be as a single unit each, so consider each single unit of different subject books as a single object. Therefore, there are
Hence, the total number of arrangements possible
Exercise 7: A student has to answer
Ans : Given: Total number of questions in part A
Total number of questions in part B
Number of questions to be answered
Minimum number of questions that must be attempted from each part
Consider different cases for the number of questions he can do from the two parts such that the sum of the attempted number of questions is
At least
(1) From part A he can attempt
Therefore, number of possible ways
(2) From part A he can attempt
Therefore, number of possible ways
(3) From part A he can attempt
Therefore, number of possible ways
Hence, the total number of ways to choose questions
Example 8: Suppose
Ans : Given: Total number of men
Total number of women
No two women are to be seated together.
First arrange all the men and then arrange all the women between them such that any two women are not adjacent to each other.
First let us arrange all the men.
_ M _ M _ M _ M _ M _ M … M _ (total
Now, when the number of men is
Also,
Therefore, the total number of arrangements
Example 9: Three married couples are to be seated in a row having six seats in a cinema hall. If spouses are to be seated next to each other, in how many ways can they be seated? Find also the number of ways of their seating if all the ladies sit together.
Ans : Given: Total number of married couples
Total number of seats
For (i) consider that the three pairs can be arranged among themselves in
For (ii) select the combination of seats such that the three ladies will always be sitting together and arrange them among themselves. Now, arrange the three men at three places.
Let us assume the couples as
(i) When the spouses will be sitting next to each other then there will be three pairs to be arranged at six seats. So the number of ways to arrange the couples is
Therefore, the total number of arrangements
(ii) In case all the ladies will be sitting together, let us assume the seats are numbered as
The three ladies can be arranged among themselves in
Therefore, the total number of arrangements
Example 10: In a small village, there are
Ans : Given: Total number of families in the village
Number of families having at most
Number of families to be chosen for assistance
Consider different case: -
In the first case assume that
In the second case assume that
In the third case assume that all the
Since, at least
(1)
In this case
Therefore, the total number of ways to make the choice
(2)
In this case
Therefore, the total number of ways to make the choice
(3) All the
In this case all the
Therefore, the total number of ways to make the choice
Hence, the total number of ways to make the choice
Example 11: A boy has
Ans : Given: Total number of books of interest
Number of library tickets
Consider three cases: -
In the first case assume that Mathematics Part I is not borrowed.
In the second case assume that both Mathematics Part I and II are borrowed.
In the third case assume that Mathematics Part I is borrowed but not II.
The following three cases are possible: -
(1) Mathematics Part I is not borrowed.
So, in this case Mathematics Part II will also not be borrowed. That means the three books can be borrowed from the remaining
(2) Both Mathematics Part I and II are borrowed.
So, in this case the remaining one book can be borrowed from the remaining
(3) Mathematics Part I is borrowed but II is not borrowed.
So, in this case the remaining two books can be borrowed from the remaining
Hence, the total number of ways to borrow the books
Example 12: Find the number of permutations of
Ans : Given: Total number of distinct things
Number of things selected
First select
Since from a total of
Therefore, now
The
Now, these
Hence, the total number of permutations
Example 13: There are four bus routes between A and B; and three bus routes between B and C. A man can travel round – trip in number of ways by bus from A to C via B. If he does not want to use a bus route more than once, in how many ways can he make round trip?
(A)
(B)
(C)
(D)
Ans : The correct option is A.
Given: Total number of bus routes from A to B
Total number of bus routes from B to C
First choose one of the paths from A to B then choose one of the paths from B to C. Now, in returning choose a different path from C to B and then from B to A.
Since, there are four routes from A to B, there are
On returning from C to B a route is to be chosen other than the route chosen to travel from B to C. So there are two routes left and one of them can be chosen in
Hence, the total number of ways to make a round – trip
Correct Answer: A
Example 14: In how many ways a committee consisting of
(A)
(B)
(C)
(D)
Ans : The correct option is B.
Given: Total number of men
Total number women
Number of men to be chosen
Number of women to be chosen
Use the formula
To choose
Therefore, the total number of ways to form the committee
Correct Answer: B
Example 15: All the letters of the word ‘EAMCOT’ are arranged in different possible ways. The number of such arrangements in which no two vowels are adjacent to each other is
(A)
(B)
(C)
(D)
Ans: The correct option is B.
Given: The word ‘EAMCOT’.
First arrange all the consonants of the word and then arrange the vowels between them such that any two vowels are not side by side.
There are three vowels (A, E, O) and three consonants (C, M, T) in the word ‘EAMCOT’. Let us first arrange the consonants.
_ C _ M _ T _
Now, the three consonants can be arranged among themselves in
Hence, the total number of ways
Correct Answer: B
Example 16: Ten different letters of the alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is
(A)
(B)
(C)
(D)
Ans : The correct option is A.
Given: Total number of letters given
Number of letters to be chosen
Find the number of five letter words that can be formed if repetitions of letters are allowed. Now, find the number of five letter words that can be formed by five different letters given.
Consider the difference of the number of words formed in the two conditions.
Since there are
In case the letters are allowed to repeat then each box can be filled in
If none of the letters are repeated then the five letters can be selected and arranged in five boxes in
Therefore, the number of words formed which have at least one letter repeated will be the difference of the number of words formed if the letters can be repeated and the number of words formed if no letter is repeated.
Hence, total number of words formed
Correct Answer: A
Example 17: The number of signals that can be sent by
(A)
(B)
(C)
(D)
Ans : The correct option is B.
Given: Total number of flags of different colours
Consider different cases where the number of flags used to send the signals will vary. For more than one flag taken at a time consider their order of sending the signals also.
One or more flags are being used to send the signals so the following cases arise: -
(1) Only one flag is used.
So, one flag can be selected in
(2) Two flags are used.
So, two flags can be selected in
(3) Three flags are used.
The total number of ways will be
(4) Four flags are used.
The total number of ways will be
(5) Five flags are used.
The total number of ways will be
(6) All the six flags are used.
The total number of ways will be
Hence, the total number of ways to send the signals
Correct Answer: B
Example 18: In an examination there are three multiple choice questions and each question has
(A)
(B)
(C)
(D)
Ans : The correct option is D.
Given: Number of multiple choice questions
Number of choices in each question
Find the number of ways he can answer all the questions. Now, subtract
Each of the three multiple choice questions has
Since, only one choice among the four choices in each of the questions will be correct, there will be only a
Therefore, the number of ways in which the student can fail to get all answer correct
Correct Answer: D
Example 19: The straight lines
(A)
(B)
(C)
(D)
Ans : The correct option is B.
Given: Total number parallel lines
Number of points on line
Number of points on line
Number of points on line
Find the total number of points on the three lines and select any three of them. Subtract the number of cases where all the points lie on the same line.
Three no – collinear points are required to form a triangle. Since the total number of points on the three lines are
Now, in case all the points lie on the same line then they will not form a triangle. So, if all the three points are chosen from amongst the
Similarly,
Hence, the total number of triangles formed
Correct Answer: B
Exercise
Short Answer Type Questions:
1: Eight chairs are numbered
Ans: Given: Number of chairs
Number of women
Number of chairs
Women choose the chairs from amongst the chairs
Use the formula
First
Now,
Since both the activities are to be performed, so the total number of possible arrangements
2: If the letters of the word RACHIT are arranged in all possible ways as listed in the dictionary. Then what is the rank of the word RACHIT?
Ans: Given: The word RACHIT.
Find the number of words that can be formed with the starting letters as A, C, H and I. Take the sum of the number of words formed in each case and add
In the word RACHIT there are
Total number of words that can be formed with the starting letter as A
Total number of words that can be formed with the starting letter as C
Total number of words that can be formed with the starting letter as H
Total number of words that can be formed with the starting letter as I
Now, the next word will be RACHIT, so its rank
3: A candidate is required to answer
Ans: Given: Total number of questions
Number of questions to be answered
Number of questions in the
Maximum number of questions that can be attempted from each group
Consider different cases for the number of questions he can do from the two groups such that the sum of the attempted number of questions is
Since the candidate has to attempt
(1) From the first group he can attempt
Therefore, number of possible ways
(2) From the first group he can attempt
Therefore, number of possible ways
(3) From the first group he can attempt
Therefore, number of possible ways
(4) From the first group he can attempt
Therefore, number of possible ways
Hence, the total number of ways to attempt questions
4: Out of
Ans: Given: Total number of points in a plane
Number of collinear points
Use the formula
Use the formula
Subtract
To form a straight line segment
Since
Therefore, the total number of distinct straight lines that will be formed by the given
5: We wish to select
Ans: Given: Total number of persons
Number of persons to be selected
If person A is chosen then person B must be chosen.
Consider three cases: -
In the first case assume that A is selected and find the possible ways.
In the second case consider that A is not selected and find the number of ways.
In the third case consider that B is selected but A is not and find the number of ways.
Add the number of ways obtained in all the cases to get the answer.
Since,
Case (1): - Considering that person A is selected. That means person B is also selected.
Therefore, number of ways to select the remaining
Case (2): - Considering that person A is not selected. That means person B is also not selected.
Therefore, number of ways to select the remaining
Case (3): - It may also be possible that B is selected and not A because there is no such given condition that if B is selected then A must also be selected.
Therefore, number of ways to select the remaining
Hence, the total number of ways of selection
6: How many committees of five persons with a chairperson can be selected from
Ans: Given: Total number of persons
Number of persons to be selected
Number of chairperson
Use the formula
Since
Now,
Hence, the total number of ways to form the committee
7: How many automobile license plates can be made if each plate contains two different letters followed by three different digits?
Ans: Given: Count of different letters on the number plate
Count of different digits on the number plate
Use the formula
First two different letters are to be selected and arranged among themselves and there are
Now, three different digits are to be selected and arranged among themselves and there are
Hence, the total number of license plates that can be made
8: A bag contains
Ans: Given: Total number of black balls
Total number of red balls
Number of black balls to be selected
Number of red balls to be selected
Use the formula
Since there are
Now, there are
Hence, the total number of ways in which
9: Find the number of permutations of
Ans: Given: Total number of distinct things
Number of things selected
First select
Since from a total of
Therefore, now
The
Now, these
Hence, the total number of permutations
10: Find the number of different words that can be formed from the letters of the word ‘TRIANGLE’ so that no vowels are together.
Ans: Given: The letters of the word ‘TRIANGLE’.
First arrange all the consonants of the given word and then arrange the vowels between them such that two vowels must not occur adjacently.
In the word ‘TRIANGLE’, the three vowels are A, E, I and the five consonants are G, L, N, R, T.
_ G _ L _ N _ R _ T _
Now, the
So the total number of arrangements
Hence, the total number of words that can be formed
11: Find the number of possible integers greater than
Ans: Given: Numbers
Consider two cases: -
In the first case fill the unit place digit as
In the second case fill the unit place digit as
Add the total numbers of ways obtained in the two cases.
Since the integer is to lie between
(1) The unit's place digit is
(2) The unit's place digit is
Hence, the total number of integers that can be formed
12: There are
Ans : Given: Total number of persons
Number of persons to be selected
Use the formula
Since
Therefore, the number of ways to select
Now,
Therefore, total number of arrangements
13: There are
Ans : Given: Total number of lamps
Their switches are independent of each other.
Use the formula
Since there are
Now, one of the combinations will be such that all the switches are off and therefore the hall cannot be illuminated. To illuminate the hall at least one of the switches must be on.
Therefore, the total number of ways to illuminate the hall
14: A box contains two white, three black and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw?
Ans : Given: Total number of white balls
Total number of black balls
Total number of red balls
Number of balls to be drawn
Consider three cases: -
In the first case, draw one black ball and find the ways to draw the other two balls.
In the second case draw two black balls and find the ways to draw the remaining one ball.
In the third case draw all the three black balls.
Since at least one black ball is to be drawn, so the following three cases arise: -
(1) One black ball is drawn from a total of three black balls and the other two balls are drawn from the remaining six balls.
Therefore, the number of ways to select
(2) Two black balls are drawn from a total of three black balls and the remaining one ball is drawn from the remaining six balls.
Therefore, the number of ways to select
(3) All the three black balls are drawn.
Therefore, the number of ways to select
Hence, the total number of ways to draw at least one black ball
15: If
Ans : Given:
Use the formula
Using the formulas
Similarly,
Solving equations
Therefore, the value of
16: Find the number of integers greater than
Ans: Given: The digits
The integer
Consider two cases: -
In the first case form all the five digit integers using the formula
In the second case form all the four digits integers that are greater than
Add the number of arrangements of the two cases.
Since the number to be formed should be greater than
(1) All the five digit integers will be greater than
(2) The four digit integers in which the thousands place digit is either
Now, the other three places can be filled (with the remaining four digits) in
Therefore, the number of four digit integers which are greater than
Hence, in overall the number of integers greater than
17: If
Ans: Given: Number of lines
No two lines are parallel and no three lines are concurrent (meet at the same point).
Use the formula
Since two non – parallel lines intersect at only one point so the total number of points of intersection of
Therefore, the number of ways to select
18: If a certain city, all telephone numbers have six digits, the first two digits always being
Ans: Given: Number of digits in a telephone number
The first two digits are either
Use the formula
There are
Therefore, the number of ways to form the telephone numbers
19: If an examination, a student has to answer
Ans: Given: Total number of questions
Number of questions to be answered
Questions
Use the formula
Since two out of five questions are compulsory and the student has to do any four questions. That means the remaining two questions are to be selected from questions
Therefore, the number of ways to select any two questions out of the remaining three
Hence, the number of ways in which the student can make the choice is
20: A convex polygon has
Ans : Given: Total number of diagonals of the convex polygon
Use the formula, number of diagonals
The diagonals of an
An
Therefore, the number of diagonals
Using the middle term split method,
Here
21: 18 mice were placed in two experimental groups and one control group, with all groups equally large. In how many ways can the mice be placed into three groups?
Ans : Given: Total number of mice
Number of experimental groups
Number of control group
All the groups are equally large.
Use the formula
Multiply both the expressions.
Since all groups are equally large that means they will contain
The remaining
Now, among
Since nothing is given that the two experimental groups are different so they are assumed to be similar. Therefore, there is no need to select any one of the two experimental groups to place
Hence, the total number of ways
22: A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if
(a) they can be of any colour
Ans: Given: Total number of white marbles
Total number of red marbles
Number of marbles drawn
use the formula
The number of ways to select any
(b) two must be white and two red and
Ans: Given: Total number of white marbles
Total number of red marbles
Number of marbles drawn
use the formula
Two white marbles can be selected out of a total of six white marbles in
Therefore, the total number of ways of selection of two white and two red marbles
(c) they must all be of the same colour.
Ans: Given: Total number of white marbles
Total number of red marbles
Number of marbles drawn
consider two cases. In the first case assume all the selected marbles are white and in the second case assume all are red. Add the number of ways obtained in the two cases.
To select all the marbles of the same colour, there can be two cases: -
(i) All the marbles are white, so the number of ways of selection
(i) All the marbles are red, so the number of ways of selection
Therefore, the total number of ways of selection of marbles (all of the same colour)
23: In how many ways can a football team of
(i) Include
Ans: Given: Total number of players
Number of players to be selected
To select any
use the formula
The total number of ways to select any
Since
Therefore, the number of ways of selection
(ii) Exclude
Ans: Given: Total number of players
Number of players to be selected
To select any
use the formula
The total number of ways to select any
Since
Therefore, the number of ways of selection
24: A sports team of
Ans : Given: Total number of students in class XI and XII each
Number of students in the sports team
At least
Consider two cases: -
In the first case consider that
In the second case consider that
Since
(1)
(2)
Hence, the total number of ways of forming the team
25: A group consists of
(i) No girls.
Ans: Given: Total number of girls in the group
Total number of boys in the group
Number of persons to be selected
use the formula
Here all the members in the team should be a boy, so the total number of ways of selection
(ii) At least one boy and one girl.
Ans: Given: Total number of girls in the group
Total number of boys in the group
Number of persons to be selected
consider different cases where there is at least one boy and one girl present in the team in each case.
Here at least one boy and one girl should be present in the team, so the following cases can be considered: -
(a) One boy and four girls are selected, so the number of ways
(b) Two boys and three girls are selected, so the number of ways
(c) Three boys and two girls are selected, so the number of ways
(d) Four boys and one girl are selected, so the number of ways
Hence, the total number of ways of selection of the team
(iii) At least three girls.
Ans: Given: Total number of girls in the group
Total number of boys in the group
Number of persons to be selected
consider different cases where there is a minimum of three girls in the team in each case.
Here at least three girls should be present in the team, so the following cases can be considered: -
(a) Two boys and three girls are selected, so the number of ways
(b) One boy and four girls are selected, so the number of ways
Hence, the total number of ways of selection of the team
26: If
(A)
(B)
(C)
(D)
Ans: The correct answer is A.
Given: The expression
Use the formula
The given expression is
Comparing the expression on both the sides,
Correct Answer: A
27: The number of possible outcomes when a coin is tossed
(A)
(B)
(C)
(D)
Ans: The correct answer is B.
The correct answer is option B.
The possible outcomes when a coin is tossed one time are
When a coin is tossed
Therefore, the total number of possible outcomes is
Correct Answer: B
28: The number of different four digit numbers that can be formed with the digits
(A)
(B)
(C)
(D)
Ans: The correct answer is C.
Given: The digits
Consider four boxes where four digits are to be filled with four different digits and find the number of ways to fill each box. Multiply all the number of ways.
Let us consider that there are four boxes and each box is to be filled with the digits
The first box can be filled in
Therefore, the total number of different four digit numbers that can be formed
Correct Answer: C
29: The sum of the digits in unit place of all the numbers found with the help of
(A)
(B)
(C)
(D)
Ans : The correct answer is B.
Given: The digits
Consider different cases where fix the unit place digit and find the number of ways to form different numbers. Take the sum of the unit place digits of all the numbers in all the cases.
The provided digits are
(1) When the unit place digit is fixed to be
Since the unit place of each digit is
(2) When the unit place digit is fixed to be
Therefore the sum of the unit place digits
(3) When the unit place digit is fixed to be
Therefore the sum of the unit place digits
(4) When the unit place digit is fixed to be
Therefore the sum of the unit place digits
Hence, the sum of unit digits of all the different numbers formed
Correct Answer: B
30: The number of words formed by
(A)
(B)
(C)
(D)
Ans: The correct answer is C.
Given: Total number of vowels
Total number of consonants
Number of vowels to be selected
Number of consonants to be selected
Use the formula
Now, arrange the
Since
Also,
Now, the
Hence, the total number of different words that can be formed
Correct Answer: C
31: A five digit number divisible by
(A)
(B)
(C)
(D)
Ans : The correct answer is A.
Given: The digits
Consider different cases where the sum of the selected digits is divisible by
Add the total number of five digit numbers obtained in all cases.
For a number to be divisible by
-
(1) The digits selected are
So the leftmost digit can be chosen amongst the four digits
Therefore, the total number of five digit numbers
(2) The digits selected are
Hence, the total number of five digit numbers (without repetition), divisible by
Correct Answer: A
32: Everybody in a room shakes hands with everybody else. The total number of handshakes is
(A)
(B)
(C)
(D)
Ans: The correct answer is B.
Given: Total number of handshakes
Assume the total number of persons as
For a handshake to a person there must be two persons, so the total number of handshakes will be equal to the total number of ways of selecting any two persons.
Let us assume the total number of persons in the room is
Using the middle term split method,
Since the total number of persons cannot be negative so
Correct Answer: B
33: The number of triangles that are formed by choosing the vertices from a set of
(A)
(B)
(C)
(D)
Ans : The correct answer is D.
Given: Total number of points
Total number of collinear points
Form three cases: -
In the first case assume that all the three vertices are taken from the set of non – collinear points.
In the second case assume that two vertices are taken from the set of non – collinear points and one vertex from the set of collinear points.
In the third case assume that one vertex is taken from the set of non – collinear points and two vertices from the set of collinear points.
There are
(1) All the three vertices are taken from the set of non – collinear points, so the number of triangles
(2) Two vertices are taken from the set of non – collinear points and one vertex from the set of collinear points. So the number of triangles
(3) One vertex is taken from the set of non – collinear points and two vertices from the set of collinear points. So the number of triangles
Hence, the total number of triangles that can be formed
Correct Answer: D
34: The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is
(A)
(B)
(C)
(D)
Ans : The correct answer is D.
Given: Set of four parallel lines intersecting another set of three parallel lines.
Select two parallel lines from each set of intersecting parallel lines and multiply the number of selections.
To form a parallelogram two sets of parallel intersecting another two sets of parallel lines is required. So the total number of parallelograms that can be formed will be equal to the number of ways to select two parallel lines from each set.
From the set of four parallel lines two lines can be selected in
From the set of three parallel lines two lines can be selected in
Therefore, the number of parallelograms that can be formed
Correct Answer: D
35: The number of ways in which a team of eleven players can be selected from
(A)
(B)
(C)
(D)
Ans : The correct answer is C.
Given: Total number of players available
Number of payers to be selected
Number of players always included
Number of players always excluded
Use the formula
Since there are
Therefore, the total number of ways of selection of the team
Correct Answer: C
36: The number of
(A)
(B)
(C)
(D)
Ans : The correct answer is D.
Given: A telephone number made up of five digits.
Consider that all the digits can be repeated and find the number of telephone numbers. Now, assume that no digit can be repeated and find the number of telephone numbers that can be formed.
Subtract the number obtained in the later assumption from the former one.
There are only five digits in the telephone number. In case all the digits can be repeated then the number of telephone numbers that can be formed is
If no digits are repeated then the number of telephone numbers that can be formed
Therefore, the number of telephone numbers having at least one of its digit repeated
Correct Answer: D
37: The number of ways in which we can choose a committee from four men and six women so that the committee includes at least two men and exactly twice as many women as men is
(A)
(B)
(C)
(D) None
Ans: The correct answer is A.
Given: Total number of men
Total number of women
Consider two different cases: -
In the first case assume there are two men and four women.
In the first case assume there are three men and six women.
The number of men should be at least two and the number of women must be twice the number of men. So the following cases arise: -
(1) There are two men (out of four) and four women (out of six) selected. Therefore, the number of ways to select them
(2) There are three men (out of four) and six women (out of six) selected. Therefore, the number of ways to select them
Hence, the number of ways to form the committee
Correct Answer: A
38: The total number of
(A)
(B)
(C)
(D)
Ans : The correct answer is C.
Given: The
Consider that there are nine boxes to be filled with nine different digits. Using the condition that the leftmost place cannot be filled with the digit
Since
Now, the leftmost place (box) cannot be filled with
After the leftmost box is filled we still have
Hence, the total number of
Correct Answer: C
39: The number of words which can be formed out of the letters of the word ARTICLE, so that vowels occupy the even place is
(A)
(B)
(C)
(D)
Ans : The correct answer is B.
Given: The word ARTICLE.
First arrange all the consonants of the given word. Now, arrange the vowels between these consonants.
There are three vowels (A, E, I) and four consonants (C, L, R, T) in the word article. Since vowels have to occupy the even places so the word will start with a consonant.
Seven places are to be filled and at positions
Now, the positions
Hence, the total number of words that can be formed
Correct Answer: B
40: Given
(A)
(B)
(C)
(D)
Ans : The correct answer is B.
Given: Total number of green dyes
Total number of blue dyes
Total number of red dyes
Consider
Since there are
Number of ways of choosing
Number of ways of choosing
Number of ways of choosing
At least one green and one blue dye are to be selected. The number of ways in which no green dye will be selected will be
Therefore, the number of ways in which at least one green dye is selected will be
Hence, the number of combinations of dyes that can be chosen
Correct Answer: B
Fill in the Blanks in Exercises
41: If
Ans : Given:
Divide
Dividing
Using the formulas
On comparing both the sides,
42:
Ans : Given: The expression
Take the terms
The provided expression is
Using the formula
Now, using the relation
43: The number of permutations of
Ans : Given: Total number of objects
Number of objects taken at a time
Repetitions are allowed.
The total number of permutations of
Think that there
Therefore, the total number of permutations will be the product
Hence, the total number of permutations is
44: The number of different words that can be formed from the letters of the word INTERMEDIATE such that two vowels never come together is _____.
Ans : Given: The word INTERMEDIATE.
First arrange all the consonants and then place the vowels between them such that no two vowels are placed adjacent to each other.
The consonants in the word INTERMEDIATE are D, M, N, R and T where T is repeated twice. Now, the vowels are A, E and I where E is repeated thrice and I is repeated twice.
First arranging the consonants,
_ D _ M _ N _ R _ T _ T _
The six consonants can be arranged among themselves in
Hence, the total number of words that can be formed
45: Three balls are drawn from a bag containing
Ans : Given: Total number of red balls
Total number of white balls
Total number of black balls
Number of balls drawn
Consider two different cases: -
In the first case assume two red balls are drawn and the remaining one ball drawn is of any colour (white or black).
In the second case assume that all the three balls drawn are red.
Since three balls are drawn in which at least two balls will be red. So the following two cases arise: -
(1) Two balls drawn are red and one ball drawn is either white or black. There are
Therefore, the number of ways to draw the balls
(2) All the three balls drawn are red.
Therefore, the number of ways to draw the balls
Hence, the total number of ways to draw at least two red balls
46: The number of six digit numbers, all digits of which are odd is _____.
Ans : Given: A six digit number is to be formed and the digits are to be odd.
Assume that there are six boxes to be filled with the digits
Since a six digit number is to be formed whose all the digits should be odd that means the choices are
Suppose there are six boxes to be filled with
Hence, the total number of six digit numbers that can be formed is
47: In a football championship,
Ans : Given: Total number of matches played
Assume that there are
Let us assume that there are
For a match to occur there must be two teams, so the total number of matches played will be equal to the number of ways in which any two teams can be selected which is given as
Using the middle term split method,
Here
48: The total number of ways in which six
Ans : Given: Total number of
Total number of
First arrange all the
Since no two
_
As it can be seen that there are seven blank spaces left where
The total number of selections of four spaces out of seven
49: A committee of
Ans: Given: Total number of men
Total number of women
Number of member to be chosen
Minimum number of men
Minimum number of women
Consider two cases: -
In the first case assume that
In the second case assume that
Since at least
(1)
(a) It may be possible that those two women are not selected, so the number of ways to select the three women out of the remaining five is
(b) One of them is selected in
So, the overall ways to form the committee for case (1)
(2)
(a) Those two women are not selected, so the number of ways to select the two women out of the remaining five is
(b) One of them is selected in
So, the overall ways to form the committee for case (2)
Hence, the total number of ways to form the committee
50: A box contains
Ans : Given: Total number of white balls
Total number of black balls
Total number of red balls
Number of balls drawn
Consider three different cases: -
In the first case assume one black ball is drawn and the remaining two balls drawn are of any colour (white or red).
In the second case assume two black balls are drawn and the remaining one ball drawn is of any colour (white or red).
In the third case assume that all the three balls drawn are black.
Since three balls are drawn in which at least one will be black. So the following three cases arise: -
(1) One ball drawn is black and the remaining two balls drawn are either white or red. There are
Therefore, the number of ways to draw the balls
(2) Two balls drawn are black and the remaining two balls drawn are either white or red.
Therefore, the number of ways to draw the balls
(3) All the three balls drawn are black.
Therefore, the number of ways to draw the balls
Hence, the total number of ways to draw at least one black ball
State whether the statements in Exercises from
51: There are
Ans : The given statement is false.
To form a straight line
Since
Therefore, the total number of distinct straight lines that will be formed by the given
52: Three letters can be posted in five letterboxes in
Ans : The given statement is false.
Here it is possible that all the three letters can be posted in the same letterbox. So there are
Similarly, there are
53: In the permutations of
Ans : The given statement is false.
Since from a total of
Therefore, now
The
Now, these
Hence, the total number of permutations
54: In a steamer there are stalls for
Ans : The given statement is true.
It can be assumed that there are
All the animals cannot be filled in one stall. So the total number of ways to fill the
55: If some or all of
Ans : The given statement is true.
There are two choices for each object, either it gets selected or not selected. So the total number of combinations for
Now, one combination will be such that none of the objects got selected, so the total number of ways to select some or all of
56: There will be only
Ans : The given statement is true.
Since at least one red ball is to be selected that means either
Now, nothing is given about the black balls, so it may be possible that
Hence, the total number of ways
57: Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side of the table. The number of ways in which the seating arrangements can be made is
Ans: The given statement is true.
Since
Therefore, the number of arrangements
Exercise 58: A candidate is required to answer
Ans : The given statement is false.
The following cases for attempting the questions from the two groups are possible: -
(1)
The number of ways in which one of the groups can be selected is
Therefore, total number of ways
(2)
Here, the number of ways to select one of the groups is
Therefore, total number of ways
Hence, the total number of ways
59: To fill
Ans : The given statement is true.
Since
Now, for the remaining
Hence, the total number of ways
In each of the Exercises from
60: There are
(a) One book of each subject : | (i) |
(b) At least one book of each subject : | (ii) |
(c) At least on book of English : | (iii) |
Ans: (a) Any one of the three Mathematics books, any one of the four Physics books and any one of the five English books can be selected. So, one book of each subject can be selected in
(b) There are
So, there are
Therefore, the total number of ways to select at least one book of each subject
(c) Here Mathematics and Physics books may or may not be selected but at least one English book is to be selected, so the total number of ways
Hence, the correct match is (a) – (ii), (b) – (iii), (c) – (i).
61: Five boys and five girls form a line. Find the number of ways of making the seating arrangement under the following condition: -
(a) Boys and girls alternate : | (i) |
(b) No two girls sit together : | (ii) |
(c) All the girls sit together : | (iii) |
(d) All the girls are never together : | (iv) |
Ans : (a) Let us first arrange the five boys.
_ B _ B _ B _ B _ B _
Since the boys and girls are to be alternate, either the first five positions will be filled or the last five positions. So there are
Therefore, the total number of arrangements
(b) In this condition no two girls sit together that means the five girls can be arranged at any
Therefore, the total number of arrangements
(c) All the five girls must sit together, which means they can be considered as one unit. They can be arranged among themselves in
Therefore, the total number of arrangements
(d) The number of arrangements where all the girls are never together will be the difference between the total number of arrangements (without any condition) and the number of arrangements where all the girls are together.
Since there are
Therefore, the number of arrangements where all the girls are never together
Hence, the correct match is (a) – (iii), (b) – (i), (c) – (i), (d) – (ii).
62: There are
(a) In how many ways committee can be formed | (i) |
(b) In how many ways a particular professor is included | (ii) |
(c) In how many ways a particular lecturer is included | (iii) |
(d) In how many ways a particular lecturer is excluded | (iv) |
Ans: (a)
Therefore, the total number of ways
(b)
Therefore, the total number of ways
(c)
Therefore, the total number of ways
(d)
Therefore, the total number of ways
Hence, the correct match is (a) – (iv), (b) – (iii), (c) – (ii), (d) – (i).
63: Using the digits
(a) how many numbers are formed? | (i) |
(b) how many numbers are exactly divisible by | (ii) |
(c) how many numbers are exactly divisible by | (iii) |
(d) how many numbers are exactly divisible by | (iv) |
Ans: (a)
(b) The numbers that are divisible by
Therefore, the total number of ways
(c) The numbers that are divisible by
Therefore, the total number of ways
(d) The numbers that are divisible by
Therefore, the total number of ways
Hence, the correct match is (a) – (i), (b) – (iii), (c) – (iv), (d) – (ii).
64: How many words (with or without dictionary meaning) can be made from the letters of the word MONDAY, assuming that no letter is repeated, if: -
(a) | (i) |
(b) All letters are used at a time | (ii) |
(c) At letters are used but the first is a vowel | (iii) |
Ans: (a) There are
(b)
(c) The vowels in the given word are A and O, so there are two choices to fill the first letter. Now, the remaining
Hence, the correct match is (a) – (iii), (b) – (i), (c) – (ii).
Learn Chapter 7 with Vedantu
The chapter 7 of Class 11 Maths discusses the concepts of permutation and combination. The topics included in the chapter are easy but it is only with proper guidance that students can ace the questions asked in the exam from this chapter. Vedantu provides exemplary solutions for the same reason.
The aim of the Vedantu's experts is to keep the students at ease and help them comprehend the concepts of combinations and permutation, the properties, and a few other crucial formulas. The provided solutions are not only useful in clearing the doubts of the students but they are also able to prepare the subject more efficiently.
Vedantu is ready to help students learn permutations of both types with repetition and without repetition. Similarly, you will get a detailed guide for both types of combinations, with repetition and without repetition.
FAQs on NCERT Exemplar for Class 11 Maths Chapter 7 - Permutations and Combinations (Book Solutions)
1. How to search for the integers number which is greater than 7000? Also, this could be formed with the digits 5, 3, 8, 7, and 9? Here any digit does not repeat!
The integers number, which is greater than 7000 could be solved in the following way: We need to ensure that we can create a maximum 5 digit number as we are not allowed to repeat any digit. Because all the provided 5 digits are greater than 7000, so we will get a 5 digit integers number that will be = 5x4x3x2x1 = 120. Similarly, a 4 digit integer will be greater than 7000 in case a thousand places contains any one of 9, 7, or 8. Hence, the place of the thousandth could be filled in 3 different ways. The three places that remained could be filled with the other four digits in a 4P3 manner. Thus, the total of four digit integers can be calculated as 3x 4P3 = 3x4x3x2 = 72 Total number of integers = 120 + 72 = 192.
2. What topics am I going to learn in NCERT Exemplar for Class 11 Math - Permutations and Combinations?
In the chapter, NCERT Exemplar for Class 11 Math - Permutations and Combinations, you are going to learn four different topics. The first one 7.1 is all about the introduction of permutations and combinations, the second one is 7.2 that comprises fundamental principles of counting, the third one 7.3 defines permutations and related aspects, and the last one 7.4 explains everything about combinations. So, this is all about the topics covered in this specific chapter of class 11 mathematics.
3. How to be exam ready while preparing NCERT Exemplar for Class 11 Math - Permutations and Combinations?
If you are preparing chapter 7 of NCERT Exemplar for Class 11 Math - Permutations and Combinations, then it takes a few steps to be fully exam ready. Your first step while preparing for the subject is to learn the formulas and equations. Once your mind is clear in terms of formulas, then it is time to cover topic-wise study. Then, the next step is to plan a timetable for regular study. And, once the syllabus is complete, you should start revising all the chapters immediately.
4. Will Vedantu prove to be a complete guide for preparing NCERT Exemplar for Class 11 Math Chapter 7 - Permutations and Combinations (Book Solutions)?
You can call Vedantu a complete package for preparing NCERT Exemplar for Class 11 Math Chapter 7 - Permutations and Combinations (Book Solutions). Vedantu works on a unique approach to guide students with basic concepts and further take them to complex ones. This way, students get habitual to the syllabus gradually and consider working on their lacking points during this process. From helping you resolve the questions to the revision procedure, Vedantu experts create a foolproof strategy to ensure your success.
5. What are the key pointers to ensure a good score in NCERT Exemplar for Class 11 Math - Permutations and Combinations?
If you want to score well in NCERT Exemplar for Class 11 Math - Permutations and Combinations, then you only need to follow a few simple steps:
The first one is to download study material from the Vedantu.
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The third one is to stick to the schedule you have prepared. Ensure that you follow this routine till your exam completes and leave enough time for revision.
The fourth one is to keep time for relaxation therapies in between otherwise learning will turn monotonous and you will avoid it unconsciously.

















