Question

Everybody in the 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. $14$

Answer 1

The correct option is B

$12$

Step 1. Find the total number of persons in the room:

Total handshakes $=66$

Let there are $n$ people in the room. 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, The number of persons in the room are $\mathbf{12}$

Hence, Option ‘B’ is Correct.