There are 30 liberals and 30 conservatives. How many 8-person committees can be formed if at least one of them must be a liberal?
If we choose a liberal and then choose the rest, we would overcount, since we could switch the order in which two liberals are picked, i.e. the distribution
\displaystyle \textrm{we pick liberal 1 and liberal 2 is chosen with the rest}is counted as a separate distribution from the identical distribution
\displaystyle \textrm{we pick liberal 2 and liberal 1 is chosen with the rest.}We solve the problem via complementary counting, since we only need to consider one case. There are \binom{60}{8} ways of choosing an 8-person committee and there are \binom{30}{8} ways of choosing no liberals, so the answer is \displaystyle\binom{60}{8} - \binom{30}{8}.