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# Everybody in a room shakes hands with everybody else.

Question:

Everybody in a room shakes hands with everybody else. The total number of hand shakes is 66. The total number of persons in the room is

A. 11

B. 12
C. 13
D. 14

Solution:

B. 12

Explanation:

We know that,

nCr

$=\frac{n !}{r !(n-r) !}$

Let total time of handshakes=nC2 = 66

$\frac{n !}{2 !(n-2) !}=66$

$\Rightarrow \frac{n(n-1)}{2}=66$

⇒ n2-n=132

⇒ (n-12) (n+11) = 0

n=12 or n = – 11

Therefore, n = 12

Hence, Option (B) 12 is the correct answer.