Hello!

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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