In how many ways could the four athletes have won the six prizes if each of them won in at least one event?

Question

Four athletes Pravin, Visharath, Bhushan and Durandhar participate in $6$ athletic events. There is only one prize for winning in each event and each of them won in at least one event. In how many ways could they have won the six prizes?

Answer 1

If all six prizes are won by the four given people, then each prize has a person associated to it. A first guess would therefore be the cardinality of $$ \{1, 2, 3, 4\}^6, $$ where we renamed the people to $1, 2, 3, 4$ respectively to shorten the notation. But there are some invalid possibilities, such as $(1, 1, 1, 1, 1, 1)$ or $(1, 2, 3, 2, 3, 1)$ where there are some people (here 4) that haven't won a prize. We have to subtract these, but each invalid possibility shall be subtracted only once.

Subtracting the ones where one person wins everything is easy: There is one such entry for each person. Then we have to subtract the $\binom{2}{4}$ possibilities for two people winning, but for each there are 2 constant vectors, so that we need to remove 2. Then finally there are $\binom{3}{4}$ possibilities for only three people winning, but we have to remove the possibilities where only one or two people win.

In total, we obtain $$ 4^6 - \binom{4}{3} \left( 3^6 - \binom{3}{2} \left( 2^6 - \binom{2}{1} \right) - \binom{3}{1} \right) - \binom{4}{2} \left( 2^6 - \binom{2}{1} \right) - \binom{4}{1}. $$