The 79 interns of the Animath internship each choose an activity for the free afternoon from 5 proposed activities. We know that:

- The swimming pool was at least as popular as football
- Students went shopping in groups of 5
- At most 4 students played cards
- At most one student stayed in their room

We write next to each activity the number of students who participated in it. How many different lists could we have written?

## Combinatorial identities
We mentally associate each student who played football with a student who went to the pool, so that the students who went to the pool + football are now distributed in pairs + an excess of students at the pool. The generating function is therefore

$$
\begin{aligned}
& \underbrace{\left(1+X^{2}+X^{4}+\ldots\right)}_{\text {football-pool pairs }} \underbrace{(1+X+\ldots)}_{\text {excess pool }} \underbrace{\left(1+X^{5}+X^{10}+\ldots\right)}_{\text {shopping }} \underbrace{\left(1+X+X^{2}+X^{3}+X^{4}\right)}_{\text {cards }} \underbrace{(1+X)}_{\text {asocial }} \\
= & \frac{1}{1-X^{2}} \frac{1}{1-X} \frac{1}{1-X^{5}} \frac{1-X^{5}}{1-X}(1+X)=\frac{1}{(1-X)^{3}}
\end{aligned}
$$

Therefore, the answer is $\binom{79+3-1}{3-1}=\binom{81}{2}$.