A committee of 12 is to be formed from nine women and eight men. In how many ways can this be done if at least five women have to be included in a committee? In how many of these committees a. the women hold majority? b. the men hold majority?

Answer

6062, (i) 2702 (ii) 1008

Explanation

Given that, there are 9 women and 8 men, a committee of 12 is to be formed including at least 5 women.
This can be done in
= (5 women and 7 men) + (6 women and 6 men) + (7 women and 5 men) + (8 women and 4 men) + (9 women and 3 men) ways
total number of ways of forming committee
`= (""^(9)C_(5)*""^(8)C_(7)) + (""^(9)C_(6)*""^(8)C_(6))+(""^(9)C_(7)*""^(8)C_(5)) + (""^(9)C_(8)*""^(8)C_(4))+(""^(9)C_(9)*""^(8)C_(3))`
`= 1008 + 2352 + 2016 + 630 + 56 = 6062`
(i) The women are in majority = 2016 + 630 + 56 = 2702
(ii) The man are in majority = 1008 ways