Question

In a party there are $10$ married couples, Each person shakes hands with every person other than her or his spouse. The total number of hand shakes exchanged in that party is ____

• A. $160$
• B. $190$
• C. $180$
• D. $170$

Answer 1

The correct option is C $180$

There are $10$ married couples.
Total number of persons $=20$
Each person shakes hands with every person,
So, total number of handshakes ${=}^{20}{C}_2$
A person does not shake hand with their spouse.
So, number of handshakes between each couple $=10$
Therefore, total handshakes exchanged in party $=$ ${}^{20}{C}_2-10$
$=\frac{20!}{2!18!}-10$
$=\frac{20\times 19}{2}-10$
$=190-10$$=180$

The total number of handshakes exchanged in a party is $180$.