Question

1. At a party, everyone shook hands with everybody else. there were 66 handshakes. How many people were at the party?
A. 30
B. 14
C. 12
D. 25​

Answer 1

Answer: 12

Step-by-step explanation:

Let, there were total M number of people named $A_{1}, A_{2}, A_{3}, ... , A_{M}$.

Now, $A_{1}$ will handshake with $(M - 1)$ people. $A_{2}$ will handshake with $(M-2)$ different people ( in this case, we exclude handshaking with $A_{1}$ because it was included in earlier case) and so on.

So, total number of handshake will be $(M - 1) + (M - 2) + (M -3) + ... +1 = \frac{1 + (M-1)}{2} (M - 1) = (M - 1) \frac{M}{2}$

Now, $(M - 1) \frac{M}{2} = 66$ ⇒ $M^{2} - M = 132$  ⇒ $M^{2} - M -132 = 0$ ⇒ $(M - 12)(M + 11) = 0$  ⇒ $M = 12$ or $M = -11$

Hence, there were 12 people, as negative value can not be possible.