# How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if,

(i) 4 letters are used at a time.

(ii) All letters are used at a time.

(iii) All letters are used but the first letter is a vowel?

Last updated date: 27th Mar 2023

•

Total views: 306.9k

•

Views today: 8.83k

Answer

Verified

306.9k+ views

Hint: Here we go through by applying the properties of permutation and combination for arranging the letters to form the word. As we know if we select r out of n it can be written as ${}^n{C_r}$ and if we permute r out of n it can be written as ${}^n{P_r}$.

Complete step-by-step answer:

Here in the question the given word is MONDAY. Here we clearly see there are 6 different letters in the word MONDAY.

(i) Number of 4-letter words that can be formed from the letters of the word MONDAY, without repetition of letters, is the number of permutations of 6 different objects taken 4 at a time, which is ${}^6{P_4}$.

Thus, required number of words that can be formed using 4 letters at a time is

${}^6{P_4} = \dfrac{{6!}}{{(6 - 4)!}} = \dfrac{{6!}}{{2!}} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 6 \times 5 \times 4 \times 3 = 360$ As we know ${}^n{P_r}$ is written as$\dfrac{{n!}}{{(n - r)!}}$.

(ii) There are 6 different letters in the word MONDAY.

When all the letters are used at time then there are six places and the six words in which the first place can be filled in 6 ways.

Second place can be filled by any one of the remaining 5 letters. So, second place can be filled in 5 ways.

Third place can be filled by any one of the remaining 4 letters. So, third place can be filled in 4 ways.

So, on continuing, number of ways of filling fourth place in 3 ways, fifth place in 2 ways, six places in 1 way.

Therefore, the number of words that can be formed using all the letters of the word MONDAY, using each letter exactly once is $6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! = 720$.

Alternative method for solving this part:

Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutations of 6 different objects taken 6 at a time, which is $6!$. (As we know arranging n letters at n place is written by n!).

(iii) Total number of letters in the word MONDAY is 6, Number of vowels are 2 i.e. (O, A)

Six letters word is to be formed.

First letter should be a vowel. So, the rightmost place of the words formed can be filled only in 2 ways.

Since the letters cannot be repeated, the second place can be filled by the remaining 5 letters. So, second place can be done in 5 ways.

Similarly, third place in 4 ways, fourth place in 3 ways, fifth place in 2 ways, and sixth place in 1 way.

Hence, the required number of words that can be formed using four letters of the given word is $2 \times 5 \times 4 \times 3 \times 2 \times 1 = 240$.

Note: Whenever we face such type of question the key concept for solving the question is go through the properties of permutation and combination as we know if there are n places and n numbers are given then first place can filled by n choice then second place can be fill by (n-1) choice and so on. This property is the golden rule for solving such types of questions.

Complete step-by-step answer:

Here in the question the given word is MONDAY. Here we clearly see there are 6 different letters in the word MONDAY.

(i) Number of 4-letter words that can be formed from the letters of the word MONDAY, without repetition of letters, is the number of permutations of 6 different objects taken 4 at a time, which is ${}^6{P_4}$.

Thus, required number of words that can be formed using 4 letters at a time is

${}^6{P_4} = \dfrac{{6!}}{{(6 - 4)!}} = \dfrac{{6!}}{{2!}} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 6 \times 5 \times 4 \times 3 = 360$ As we know ${}^n{P_r}$ is written as$\dfrac{{n!}}{{(n - r)!}}$.

(ii) There are 6 different letters in the word MONDAY.

When all the letters are used at time then there are six places and the six words in which the first place can be filled in 6 ways.

Second place can be filled by any one of the remaining 5 letters. So, second place can be filled in 5 ways.

Third place can be filled by any one of the remaining 4 letters. So, third place can be filled in 4 ways.

So, on continuing, number of ways of filling fourth place in 3 ways, fifth place in 2 ways, six places in 1 way.

Therefore, the number of words that can be formed using all the letters of the word MONDAY, using each letter exactly once is $6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! = 720$.

Alternative method for solving this part:

Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutations of 6 different objects taken 6 at a time, which is $6!$. (As we know arranging n letters at n place is written by n!).

(iii) Total number of letters in the word MONDAY is 6, Number of vowels are 2 i.e. (O, A)

Six letters word is to be formed.

First letter should be a vowel. So, the rightmost place of the words formed can be filled only in 2 ways.

Since the letters cannot be repeated, the second place can be filled by the remaining 5 letters. So, second place can be done in 5 ways.

Similarly, third place in 4 ways, fourth place in 3 ways, fifth place in 2 ways, and sixth place in 1 way.

Hence, the required number of words that can be formed using four letters of the given word is $2 \times 5 \times 4 \times 3 \times 2 \times 1 = 240$.

Note: Whenever we face such type of question the key concept for solving the question is go through the properties of permutation and combination as we know if there are n places and n numbers are given then first place can filled by n choice then second place can be fill by (n-1) choice and so on. This property is the golden rule for solving such types of questions.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?