The simplest way to solve this question:
Calculate the following:
1. Number of valid teams: A valid team for a mixed doubles game implies, 1 Man and 1 Woman in the team and the 2 of them should not be married.
2. Number of games: How many games can be played among these 12 teams where there are no overlapping people while selecting the 2 teams.
Lets calculate 1:
We have 4 men and 4 women to select from.
Number of valid teams = 4*3 (each man will have 3 women teammate options - excluding his wife) = 12 teams.
Lets calculate 2:
Any team selected from the 12 teams will have 1M and 1W.
Selecting team 1:
We have 12 teams and we want to select 1 so 12C1 ways = 12 ways.
Selecting team 2:
Now the 2 people (1M & 1W) selected in team 1, cannot be a part of team 2. Any such combination would eliminate 5 teams from the 12 teams, so we are left with 7 teams, so 7C1 ways to select team 2 = 7 ways.
Total valid games = (12*7)/2 = 42 games.
Dividing by 2 because order of teams does not matter.