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