Hello, dannyshox!

Your counting is off . . .

There are 12 couples: 12 men and 12 women.A team of 7 people is to be chosen from 12 couples.

(a) How many ways can a team of 3 men and 4 women be chosen?

Choose 3 men: . ways.

Choose 4 women: . ways.

Therefore: . ways.

I found no neat formula for this . . . I made an exhaustive list.(b) How many ways can a team containing no couples be chosen?

7 Men, 0 Women

. . . ways.

6 Men, 1 Woman

Choose 6 men: . ways.

The 1 woman must be chosen from the 6 women

. . who arewives of the 6 men: . ways.not

. . ways.

5 Men, 2 Women

Choose 5 men: . ways.

The 2 women must be chosen from the 7 women

. . who arewives of the 5 men: . ways.not

. . ways.

4 Men, 3 Women

Choose 4 men: . ways.

The 3 women must be chosen from the 8 women

. . who arewives of the 4 men: . ways.not

. . ways.

3 Men, 4 Women

Choose 3 men: . ways.

The 4 women must be chosen from the 9 women

. . who arewives of the 3 men: . ways.not

. . ways.

2 Men, 5 Women

Choose 2 men: . ways.

The 5 women must be chosen from the 10 women

. . who arewives of the 2 men: . ways.not

. . ways.

1 Man, 6 Women

Choose 1 man: . ways.

The 6 women must be chosen from the 11 women

. . who arewives of the 2 men: . ways.not

. . ways.

0 Man, 7 Women

Choose 7 women: . ways.

Now add them . . .

(c) How many ways can a team containing exactly two couples be chosen?

Choose 2 of the 12 men to be on the team: . ways.

There is 1 way that their wives are also on the team.

We have 4 people on the team.: 2M, 2W.

We must choose 3 more people from the other 4 men and 4 women

. . and includecouples.no

There are 4 cases:

. .3M, 0W: .

. .2M, 1W: .

. .1M, 3W: .

. .0M, 3W: .

There are: . ways.

Therefore: . ways.