Question

After a meeting, every participant shakes hands with every other participants. If the number of handshakes is 66, find the number of participants in the meeting.

Answer 1

Let there be n participants present in the meeting.
A handshake occurs between 2 persons.
Then, the number of ways of selecting any 2 persons from them = ⁿC₂ 
Now, in all 66 handshakes were exchanged.
∴ ⁿC₂  = 66

∴ `("n"!)/(("n" - 2)!2!)` = 66

∴ `("n" xx ("n" - 1) xx ("n" - 2)!)/(("n" - 2)!)` = 66 × 2

∴ n(n – 1) = 132
∴ n(n – 1) = 12 × 11
Comparing on both sides, we get
∴ n = 12
∴ The number of participants in the meeting = 12