# Combinations problem on committee.

A committee of three boys and three girls is to be selected from a class of 14 boys and 17 girls. In how many ways can the committee be selected if: (a) Ana has to be on the committee? (b) the girls must include either Roberta or Priya, but not both?

For part a), I got the answer 43680 by doing 14C3 x 16C2 and the answer is correct but however, for part b) the answer I got is 76440 by doing 14C3 x 15C2 x 2 but the answer is wrong. Please Help!

• Who is Ana? Boy or girl? – SchrodingersCat Nov 21 '15 at 12:26
• she is a girl... – Sulaiman Muzaffer Nov 21 '15 at 12:26
• Your solution looks correct. Why do you think it's wrong? – Barry Cipra Nov 21 '15 at 12:30
• The answer at the back of the book for part b) is 65520 – Sulaiman Muzaffer Nov 21 '15 at 12:32
• @Nicholas, please take a look at my answer. If you still think inclusion-exclusion is required, post an answer showing the details. – Barry Cipra Nov 21 '15 at 13:25

Requiring just one of the two girls Roberta and Priya to be on the committee is equivalent to splitting the students into three groups: the $14$ boys, the $2$ girls Roberta and Priya, and the other $15$ girls, with the requirement that committee consist of $3$ members of the first group, $1$ member of the second group, and $2$ members of the third group. The number of choices is thus
$${14\choose3}{2\choose1}{15\choose2}=364\cdot2\cdot105=76440$$
I'm not sure how the book got the answer $65520=364\cdot180$, unless it mistakenly did some sort of inclusion-exclusion.