TITLE: Calculating number of winners possible for an archery competition.
QUESTION [0 upvotes]: There are ten contestants in an archery competition. Four of the contestants are women. Prizes are awarded to the top four competitors. If at least one woman finishes in the first four places, in how many ways can the top four places be filled?

REPLY [2 votes]: If we ignore the condition that at least one woman finishes, we would come up with $P_4^{10}=5040$ possible ways.
Now, the opposite of the condition that at least one woman finishes, is that no woman finishes, which means that the 4 spots are all not women, which gives us $P_4^6=360$, with the $6$ because there are $4$ women, hence $6$ non-women.
Therefore, the answer is $P_4^{10}-P_4^6=4680$.
From the simple Venn diagram below, one can see that:
$$\mbox{Regardless of the condition}=\mbox{Opposite of the condition}+\mbox{According to the condition}$$
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