A committee of 5 people is to be chosen from a group of 8 women and 10 men. How many diffferent committees are possible? How man
y are possible if the committee must feature both men and women? How many are possible if the committee must feature 3 women and 2 men? How many are possible if the committee must have more women than men?
A) 5 to be chosen among a Total : 10 Men + 8 Women
¹⁸C₅ = (18!)/(5!)(13!) = 8,568 groups of five
b) A must to have men and women. If so we have to deduct all groups of 5 that are all men and all group of 5 that are all women Groups of 5 with only men: ¹⁰C₅ = 252 Groups of 5 with only women: ⁸C₅ = 56
So number of committees of 5 men and women mixed = 8568 - 252 - 56 = 8,260 committees
c) 3 Women and 2 Men:
⁸C₃ x ¹⁰C₂ = 2,520 groups of 3 W and 2 M
d) More women than men, it means: 3 W + 2 M OR (we have found it in c) = 2,520) 4 W + 1 M OR ⁸C₄ x ¹⁰C₁ →→→→ = 700 5 W + 0 M OR ⁸C₅ x ¹⁰C₀ →→→→ = 56
Total where W>M = 3,276 groups of 5 where women are at least 3
