# Combinations/Permutations problem: juniors and seniors trying out for a team

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 $B$.

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.