Committee of Couples

From a group of five married couples, how many committees of four or five people can be formed if no two people on the committee may be married to each other?

4-member commitee

There are 5C4 ways to choose four couples. Then there are 2C1 ways to pick one person from each couple.

₅C₄ x ₂C₁ x ₂C₁ x ₂C₁ x ₂C₁ = 5 x 2 x 2 x 2 x 2 = 80

5-member commitee

₅C₅ x ₂C₁ x ₂C₁ x ₂C₁ x ₂C₁ x ₂C₁ = 1 x 2 x 2 x 2 x 2 x 2 = 32

The required combinations (OR = union) = 80 + 32 = 112

Without those restrictions, there could have been ₁₀C₄ + ₁₀C₅ possibilities.