8. In how many ways can a committee of four be selected from eight men and seven women if Miss Jones refuses to serve on the same committee as Mr. Smith?

If the exclusion of Miss Jones and Mr Smith serving together was not present, the are 15C4 = 1365 ways of selecting the committee.

Miss Jones can serve on the committee in the following ways:

a) with 3 men

b) with 2 men and another woman

c) with 1 man and 2 other women

d) with 3 other women.

Arrangement d) obviously presents no restrictions.

Arrangement a) has 7C3 ways excluding Mr Smith, and 8C3 ways if Mr Smith was included.

Arrangement b) has 7C2 * 6 ways excluding Mr Smith, and 8C2 * 6 ways if Mr Smith was included.

Arrangement c) has 7 * 6C2 ways excluding Mr Smith, and 8 * 6C2 ways if Mr smith was included.

The reductions in ways caused by the restriction are as follows:

a) 8C3 - 7C3 = 21 ways

b) 6 (8C2 - 7C2) = 42 ways

c) 6C2 = 15 ways

The total reduction in the number of ways is: 21 + 42 + 15 = 78.

Therefore the total number of ways of selecting the committee, while observing the restriction, is 1365 - 78 = 1287 ways.