First slide

Permutations

Question

Every body in a room shakes hands with every body else. The total number of hand shakes is 66. The total number of persons in the room is

Moderate

A

11
B

12
C

13
D

14

Solution

Let total number of persons be n.
Since two persons shake hands, so number of ways of shaking hands $=^{n} C_{2}$ .
According to question, we have $^{n} C_{2} = 66$
or $\frac{n!}{(n - 2)! 2!} = 66 or \frac{n (n - 1) (n - 2)!}{(n - 2)! 2 \times 1} = 66$
or $n^{2} - n = 132 or n^{2} - n - 132 = 0$
or $n^{2} - 12 n + 11 n - 132 = 0$
or $n (n - 12) + 11 (n - 12) = 0 \Rightarrow n = - 11 or 12$
But, n = –11 is not possible.
$∴ n = 12 .$

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