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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!

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

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.

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  • $\begingroup$ You are right, though inclusion-exclusion also gives the same answer. I'd probably misread a few numbers earlier. $\endgroup$ – Nicholas Nov 21 '15 at 13:35
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    $\begingroup$ @Nicholas, I'm glad we're on the same page. I am quite sure I've made far more mistakes than you, if only because I've had a few decades head start. With luck, you'll catch up.... $\endgroup$ – Barry Cipra Nov 21 '15 at 15:01

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