2019 Mock AMC 10B Problems/Problem 2

Problem

Al, Bob, Clayton, Derek, Ethan, and Frank are six Boy Scouts that will be split up into two groups of three Boy Scouts for a boating trip. How many ways are there to split up the six boys if the two groups are indistinguishable?

$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 35$

Solution

There are ${6 \choose 3}=20$ ways to pick three boys. However, we have overcounted by double as if we choose the latter of three boys before its former counterpart, then we get the same case, so our answer is $ \frac{20}{2} = $ $\boxed{10}$ .

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2019 Mock AMC 10B Problems/Problem 2

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Solution