A mixed doubles tennis game is to be arranged from $5$ married couples. In how many ways the game can be arranged if no husband and wife pair is included in the same game?

To arrange the game we have to do the following operations.
(i) Select two men from from $5$ men in ${}^{5}C_2$ ways

(ii) Select two women from $3$ women excluding the wives of the men already selected. This can be done in ${}^{3}C_2$ ways.
(iii) Arrange the $4$ selected persons in two teams. If the selected men are $M_1$ and $M_2$ and the selected women are $W_3$ and $W_4$ this can be done in $2$ ways:
        $M_1{W}_3$ play against $M_2{W}_4$   
        $M_2{W}_3$ play against $M_1{W}_4$
Hence the number of ways to arrange the game

$={}^{5}C_2{}^{3}C_2\left(2\right) =10\times 3\times 2=60$