Question

Find the number of ways in which we can select four numbers from 1 to 30 so as to exclude every selection of four consecutive numbers.
(A) 27378 (C) 27399
(B) 27405 (D) None of those

Answer 1

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.