Question

the total number of ways in which 4 persons can be chosen from 16 sitting in a circle such that no two of the chosen are neighbours is :

Answer 1

Given:

There are 16 persons are sitting in a circle.

To Find:

The total number of ways in which 4 persons can be chosen from 16 sitting in a circle such that no two of the chosen are neighbors is  

Solution:

Here, it is given that , 16 persons are sitting in a circle .

Therefore, Number of ways of choosing 4 persons sitting in a circle when they are neighbors is =  $^{16}C_{4}$

Now,

Number of ways when either 2 persons are neighbors = 16

Also, when exactly 2 neighbors 3rd & 4th difference = 16(2 neighbors) × 10

                                                                                 = 160

So, total number of ways of choosing 4 persons sitting in a circle such that no two of the chosen are neighbors is

$=^{16}C_{4}-160-16\\\\=1960-176\\\\=1784$

Hence, the answer is 1784.