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The problem statement from this website is as follows:

An equal number of juniors and seniors are trying out for six spots on the university debating team. If the team must consist of at least four seniors, then how many different possible debating teams can result if five juniors try out?

(A) 50 (B) 55 (C) 75 (D) 100 (E) 250

I started out by drawing 6 slots where members can be placed. I identified this as a combination problem because it does not matter whether you have a team of:

$\{Senior1, Senior2, Senior3, Senior4, Person, Person\}$

or

$\{Senior2, Senior4, Senior3, Senior1, Person, Person\}$

That is, the order doesn't matter, so we won't use permutations. They tell us that $5$ juniors try out, and by the first sentence of the problem statement, there must also be $5$ seniors trying out. We are also told that at least $4$ seniors will be trying out, which means $4/6$ of those slots are guaranteed to be filled up with seniors. This leaves $2/6$ slots that can either be filled with the remaining seniors or juniors. After meeting the minimum requirement for seniors, there is only $1$ senior remaining, so in total, there are:

  • $5$ juniors

  • $1$ senior

And there are $2$ slots. This means we have "$6$ choose $2$", which I calculated to be $15$.

They are saying the answer is $B$.

Could someone please help me understand what I did wrong?

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The problem is that you first selected four spots for senior members, but you did not specify how these seniors are chosen. To arrive at the correct answer, you can distinguish the following two scenarios:

  1. Four seniors and two juniors: ${5 \choose 4}{5 \choose 2} = 5 \cdot 10 = 50$ possibilities

  2. Five seniors and one junior: ${5 \choose 5}{5 \choose 1} = 1 \cdot 5 = 5$ possibilities

In total, we find $50 + 5 = 55$ possible arrangements.

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  • $\begingroup$ Ah, I see, so my mistake was that I assumed the seniors would all be in slots 1-4, right? Instead of doing 5 choose 4 $\endgroup$ – AleksandrH Jun 29 '17 at 21:14
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    $\begingroup$ @AleksandrH Which slots are used does not matter in this question; the problem is that you assumed that four fixed seniors were assigned to four slots already, while these seniors should be selected as well. $\endgroup$ – jvdhooft Jun 29 '17 at 21:22
  • $\begingroup$ Yep, that's what I meant. Thank you! $\endgroup$ – AleksandrH Jun 29 '17 at 21:35

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