At a meeting there are 15 people, and each person shakes hands exactly once with every other person. How many handshakes take place in all?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 105
Each handshake is fixed by a pair of people, and a pair has no order, so the count is \(^nC_2\).
Handshakes \(\displaystyle =\,^{15}C_2=\frac{15\times14}{2}=105\)
Why the other options are incorrect
Option (1) 210 — Incorrect. This is \(15\times14\); it counts every handshake twice, once from each person.
Option (2) 225 — Incorrect. This is \(15\times15\), which even lets a person shake hands with himself.
Option (4) 120 — Incorrect. This is \(^{16}C_2\), the count for 16 people.