(Q). A committee of 12 is to be formed from 9 women and 8 men.
(i). In how many ways this can be done if at least 5 women have to be included in a committee?
(ii). In how many of these committees women are in majority?
(iii). In how many of these committees the men are in majority?
Solution:-
(i).
5W7M + 6W6M + 7W5M + 8W4M + 9W3M
= ⁹C₅×⁸C₇ + ⁹C₆×⁸C₆ + ⁹C₇×⁸C₅ + ⁹C₈×⁸C₄ + ⁹C₉×⁸C₃
= 1008 + 2352 + 2016 + 630 + 56
= 6062 ways Answer
(ii).
When women are in majority then number of ways = ⁹C₇×⁸C₅ + ⁹C₈×⁸C₄ + ⁹C₉×⁸C₃
= 2016 + 630 + 56
= 2702 Answer
(iii).
When men are in majority then number of ways = ⁸C₇×⁹C₅ + ⁸C₈×⁹C₄
= 1008 + 126
= 1134 ways Answer