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. $9$
• B. $12$
• C. $10$
• D. $14$

Answer 1

The correct option is B

$12$

Step 1. Let there are $n$ people in the room.

Total handshakes $=66$

There should be two hands involved for a handshake.

$\therefore$total number of persons in a room are:

${}^{n}C_2=66$

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

$\Rightarrow$$n^2-n-132=0$

Step 2. Solve the equation for$n$, we get

$\Rightarrow$ $n^2-12n+11n-132=0$

$\Rightarrow$$n(n-12)+11(n-12)=0$

$\Rightarrow$ $(n-12)(n+11)=0$

$\therefore$$n=12$ and $n=-11$

The value of $n$ cannot be negative.

$\therefore$$n=12$

Thus, there are $\mathbf{12}$ people in the room.

Hence, Option ‘B’ is Correct.