Handshake problem: In a room, everyone shakes hands with everyone else exactly once. If the total number of handshakes is 66, how many persons are in the room?

Introduction / Context:When each pair among n people shakes hands once, the total handshakes equal the number of 2-element subsets: C(n, 2). We invert this to find n from a given total.

Given Data / Assumptions:Total handshakes = 66.

Concept / Approach:Solve C(n, 2) = 66 ⇒ n(n − 1)/2 = 66 ⇒ n(n − 1) = 132, find integer n.

Step-by-Step Solution:

Verification / Alternative check:Quadratic: n^2 − n − 132 = 0 ⇒ (n − 12)(n + 11) = 0 ⇒ n = 12.

Why Other Options Are Wrong:11 gives 55; 13 gives 78; 14 gives 91.

Common Pitfalls:Using permutations n(n − 1) instead of combinations.

Final Answer:12