At a get-together, there are 12 guests. Every guest shakes hands exactly once with every other guest. How many handshakes take place in all?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 66
A handshake takes place between two guests, and A shaking hands with B is the same handshake as B with A. So each handshake is a selection of 2 guests: \(^nC_2=\frac{n(n-1)}{2}\).
Number of guests \(n=12\)
Handshakes \(\displaystyle =\,^{12}C_2=\frac{12\times11}{2}=66\)
Why the other options are incorrect
Option (1) 132 — Incorrect. This is \(12\times11\); it counts every handshake twice (once for each of the two guests).
Option (3) 144 — Incorrect. This is \(12\times12\); a guest does not shake hands with himself, and each handshake is shared by two guests, so this count is far too large.
Option (4) 78 — Incorrect. This is \(^{13}C_2\), the count for 13 guests, not 12.