Answer
Verified
424.5k+ views
Hint: First we will find the total number of ways of selecting four numbers from 1 to 30 including selection of four consecutive numbers. Then we will find the number of ways in which we can select 4 numbers from 1 to 30 such that all the selections are four consecutive numbers. We will subtract both of them to calculate the number of ways in which we can select four numbers so as to exclude every selection of four consecutive numbers.
Complete step by step solution: First we will find total number of ways of selective four number from 1 to 30 (which also includes selection of every four consecutive number )
$\therefore $ total no. of ways of selecting four number from 1 to 30 are as follows :-
We have to select 4 numbers from 30 numbers. This can be done in
${ }^{30}{C_4}$ ways = 27405 ways.
(Reason:- The number of ways of selecting ‘r’ items from a total of ‘n’ different items is ${ }^n{C_r}$ ways)
Now we have to find the total number of ways in which we can select 4 numbers from 1 to 30 such that all the selections are of four consecutive numbers.
This case can be done in 27 ways. (Reason No. of possible selections are 27. Have is the listing of all possible selections.
1, 2, 3, 4
2, 3, 4, 5
3, 4, 5, 6
4, 5, 6, 7
have are the list of all possible selections having four consecutive numbers.
$ \Rightarrow $ 27 ways.
(25) 25, 26, 27, 28
(26) 26, 27, 28, 29
(27) 27, 28, 29, 30
So total no. of ways in which we can select four numbers from 1 to30 so as to include every selection of four consecutive number
= No. of ways in which we can select four numbers from 1 to 30 include selection of four consecutive numbers - No. of ways in which we can select only four consecutive numbers from 1 to 30.
${ }^{30}{C_4}-27 \\
\Rightarrow 21405 – 27 \\
\Rightarrow 27378\;\text{ways}$
$\therefore $ correct option - (A) 27378
Note: Here instead of directly finding the solution to this question we have used the complement method. We subtracted selections of four consecutive numbers from 1 to 30 from selection of four numbers from 1 to 30 (which include selection of four consecutive numbers ) to find selection of four numbers which exclude selection of four numbers from 1 to 30 which are consecutive .
This method is very helpful in many other questions.
Complete step by step solution: First we will find total number of ways of selective four number from 1 to 30 (which also includes selection of every four consecutive number )
$\therefore $ total no. of ways of selecting four number from 1 to 30 are as follows :-
We have to select 4 numbers from 30 numbers. This can be done in
${ }^{30}{C_4}$ ways = 27405 ways.
(Reason:- The number of ways of selecting ‘r’ items from a total of ‘n’ different items is ${ }^n{C_r}$ ways)
Now we have to find the total number of ways in which we can select 4 numbers from 1 to 30 such that all the selections are of four consecutive numbers.
This case can be done in 27 ways. (Reason No. of possible selections are 27. Have is the listing of all possible selections.
1, 2, 3, 4
2, 3, 4, 5
3, 4, 5, 6
4, 5, 6, 7
have are the list of all possible selections having four consecutive numbers.
$ \Rightarrow $ 27 ways.
(25) 25, 26, 27, 28
(26) 26, 27, 28, 29
(27) 27, 28, 29, 30
So total no. of ways in which we can select four numbers from 1 to30 so as to include every selection of four consecutive number
= No. of ways in which we can select four numbers from 1 to 30 include selection of four consecutive numbers - No. of ways in which we can select only four consecutive numbers from 1 to 30.
${ }^{30}{C_4}-27 \\
\Rightarrow 21405 – 27 \\
\Rightarrow 27378\;\text{ways}$
$\therefore $ correct option - (A) 27378
Note: Here instead of directly finding the solution to this question we have used the complement method. We subtracted selections of four consecutive numbers from 1 to 30 from selection of four numbers from 1 to 30 (which include selection of four consecutive numbers ) to find selection of four numbers which exclude selection of four numbers from 1 to 30 which are consecutive .
This method is very helpful in many other questions.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Why Are Noble Gases NonReactive class 11 chemistry CBSE
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
At which age domestication of animals started A Neolithic class 11 social science CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE