Question

In a party, there are 10 pairs arriving, each consisting of a boy and a girl. Then, there is a dance, where each boy dances with one girl (and one girl only). Each boy is equally likely to dance with each girl, and each girl is equally likely to dance with each boy. Naturally, it is likely that there are some boy-girl pairs who dance together, and who arrived together too. A squad is defined as a collection of boy-girl pairs, where boys and girls are dancing within the squad. No member in a squad can be dancing with anyone not part of that squad. Note that if a boy-girl pair is dancing together, that one single pair is also a squad on its own. Naturally, multiple such squads can form among the 10 pairs who came to the party. What is the expected number of squads

Answer 1

Expected no. of squads = 184755

Explanation:

In a party, there are 10 pairs arriving, each consisting of a boy and a girl.

So that means there are 10 boys and 10 girls.

A squad is defined as a collection of boy-girl pairs and one single pair is also a squad on its own.

We are asked to find out the the expected number of squads.

For 10 pairs of boys and girls:

Number of squads = ₁₀C₁₀ × ₁₀C₁₀ = 1 × 1 = 1

For 9 pairs of boys and girls:

Number of squads = ₁₀C₉ × ₁₀C₉ = 10 × 10 = 100

For 8 pairs of boys and girls:

Number of squads = ₁₀C₈ × ₁₀C₈ = 45 × 45 = 2025

For 7 pairs of boys and girls:

Number of squads = ₁₀C₇ × ₁₀C₇ = 120 × 120 = 14400

For 6 pairs of boys and girls:

Number of squads = ₁₀C₆ × ₁₀C₆ = 210 × 210 = 44100

For 5 pairs of boys and girls:

Number of squads = ₁₀C₅ × ₁₀C₅ = 252 × 252 = 63504

For 4 pairs of boys and girls:

Number of squads = ₁₀C₄ × ₁₀C₄ = 210 × 210 = 44100

For 3 pairs of boys and girls:

Number of squads = ₁₀C₃ × ₁₀C₃ = 120 × 120 = 14400

For 2 pairs of boys and girls:

Number of squads = ₁₀C₂ × ₁₀C₂ = 45 × 45 = 2025

For 1 pairs of boys and girls:

Number of squads = ₁₀C₁ × ₁₀C₁ = 10 × 10 = 100

The expected number of squads is

Expected no. of squads = 1 + 100 + 2025 + 14400 + 44100 + 63504 + 44100 + 14400 + 2025 + 100

Expected no. of squads = 184755