Question

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

• A. 11
• B. 12
• C. 13
• D. 14

Hint:

Number of handshakes among $n$ persons $= \dbinom{n}{2} = \dfrac{n(n-1)}{2}$. Solve the resulting quadratic for $n$.

Answer 1

The Correct Option is B

Step 1: Conceptual Understanding:
Each pair of persons shakes hands exactly once, so total handshakes $= \dbinom{n}{2}$.
Step 2: Explanation in Detail:
$\dfrac{n(n-1)}{2} = 66 \Rightarrow n(n-1) = 132 \Rightarrow n = 12$ (since $12 \times 11 = 132$).
Step 3: Therefore, Stating the Final Answer
There are $12$ persons in the room.