How many ways can 10 persons shake hands with two persons?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 45
Every handshake takes place between two persons, and the handshake of A with B is the same as that of B with A. So the number of handshakes is the number of ways of choosing 2 persons out of 10, a combination, in which order does not matter.
\(\displaystyle {}^{10}C_{2}=\frac{10\times9}{2\times1}=45\)
Tip: When each of \(n\) persons shakes hands once with every other person, the number of handshakes is \(\frac{n(n-1)}{2}\): each person shakes \(n-1\) hands, and each handshake is counted twice.