The number of ways in which a mixed doubles game can be arranged from amongst n couples such that no husband and wife play in the same game, is

A mixed doubles game involves two males and two females. Two males can be chosen from n males in nC2 ways. Having chosen two males, now 2 females are to be chosen from (n –2) females leaving the wives of the two already chosen males. This can be done in n –2C2 ways.Hence, the required number of ways=nC2⋅n−2C2⋅2        [for every choice of 2 males and 2 females, they can be paired in 2 ways].=n(n−1)2⋅(n−2)(n−3)2⋅2=n(n−1)(n−2)(n−3)2=12⋅nP4