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A reading list for a humanities course consists of 10 books; of which 4 are biographies and the rest are novels. Each student is required to read a selection of 4 books from the list including 2 or more biographies. How many selections of 4 books satisfy the requirements? (source: New Revised GRE General Guide)

I initially tried to solve the problem by doing: $\binom{4}{2} \binom{8}{2}$ (i.e., (4 choose 2) * (8 choose 2)). That gives me 168.

However, the answer is actually 115. I understand how the answer solved it: look at 3 cases:

  • Case 1: 2 bios and 2 fiction
  • Case 2: 3 bios and 1 fiction
  • Case 3: 4 bios and 0 fiction

And add up the number of combinations of each.

But what's wrong with my approach?

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Welcome to MSE =) – Patrick Da Silva Dec 28 '11 at 3:35
up vote 1 down vote accepted

Your approach counts a selection more than once if it contains more than two biographies. Then your formula wants to decide exactly which of the two biographies the biographies and the third (and possibly fourth) is considered merely an honorary novel.

For example, with biographies BA, BB, BC, BD and novels N1 through N6, your formula counts

  • {BA,BB}+{BD,N3}
  • {BA,BD}+{BB,N3}
  • {BB,BD}+{BA,N3}

as different selections even though they lead to exactly the same reading.

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Perhaps one should add that the correct approach is to say "4 choose 2" then "6 choose 2" + "4 choose 3" then "6 choose 1" + "4 choose 4" then "6 choose 0" which leads to $$ \begin{pmatrix} 4 \\ 2 \end{pmatrix}\begin{pmatrix} 6 \\ 2 \end{pmatrix} + \begin{pmatrix} 4 \\ 3 \end{pmatrix} \begin{pmatrix} 6 \\ 1 \end{pmatrix} + \begin{pmatrix} 4 \\ 4 \end{pmatrix}\begin{pmatrix} 6 \\ 0 \end{pmatrix} = 115 $$. – Patrick Da Silva Dec 28 '11 at 3:33
@Patrick, that appears to be implicit it in the part of the question beginning "I understand how the answer solved it". – Henning Makholm Dec 28 '11 at 3:34
Great! Excellent answer, your example (with the BA, BB, etc and N1 to N6) helped me see where I was double counting. Thank you so much! – oxuser Dec 28 '11 at 10:41

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