Question

Consider a group of $20$people. If everyone shakes hands with everyone else, how many handshakes take place?

Short answer

Total number of handshakes$=190$.

Step-by-step solution

Step 1. Given information.

No. of persons = $20$

Everyone shakes hands with everyone else.

Step.2 Calculate the possible number of handshakes.

No. of combinations, which can be formed out of $20$people taking $2$at a time is given by $\left(\begin{array}{c}20\\ 2\end{array}\right)$.

$$\begin{gathered} =\frac{20!}{2!18!} \\ =\frac{20\times 19\times 18!}{2\times 1\times 18!} \\ =10\times 19 \\ =190 \end{gathered}$$

Therefore, the possible no. of handshakes are$=190$.