A group of seven women and four men must select a four-person committee. How many committees are possible if it must consist of the following? (a) Two women and two men, (b) any mixture of men and women, (c) a majority of women.

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I assume that the order of people in the committee is insignificant.

(a) There are `( ( 7 ), ( 2 ) ) = ( 7 * 6 ) / ( 2 * 1 ) = 21 ` ways to select 2 women and `( ( 4 ) , ( 2 ) ) = ( 4 * 3 ) / ( 2 * 1 ) = 6 ` independent ways to select 2 men. Therefore, there are `21 * 6 = 126 ` ways to create such a committee.

(b) "Any mixture of men and women" probably means "at least onу man and at least one woman." Then, there are 3 incompatible events: 1 man, 3 women; 2 men, 2 women; 3 men, 1 woman.

The number of ways for the first possibility is `( ( 4 ) , ( 1 ) ) * ( ( 7 ) , ( 3 ) ) = 4 * 35 = 140, ` for the second it is `126 ` from (a) and for the third it is `( ( 4 ) , ( 3 ) ) * ( ( 7 ) , ( 1 ) ) = 4 * 7 = 28. ` The answer is `140 + 126 + 28 = 294 .`

(c) "A majority of women" means 0m/4w or 1m/3w, the same way as in (b) it is `( ( 7 ) , ( 4 ) ) + ( ( 4 ) , ( 1 ) ) * ( ( 7 ) , ( 3 ) ) = 35 + 4 * 35 = 175 .`

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