The 83 interns from the Animath internship each choose an activity for the free afternoon from 5 proposed activities. We know that:

$\triangleright$ Shopping has been at least as popular as Laser Game;

$\Delta$ Movie tickets are sold in lots of 6;

$\Delta$ At most 5 students go to the beach;

$\triangleright$ At most one student has visited a Romanesque church.

When writing the list of activities in alphabetical order, one writes the number of students corresponding to each activity. How many different lists of numbers could have been written?
We apply the counting technique seen in class:

$\triangleright$ For laser tag and shopping, we count the number of people who went to the Laser Game, multiply it by 2, and then count the additional number of people who went shopping. The associated series is therefore: $\frac{1}{1-x^{2}} \frac{1}{1-x}$;

$\triangleright$ for movie tickets, the generating series associated with multiples of 6 is $\frac{1}{1-x^{6}}$;

$\triangleright$ for the beach, the series is: $1+x+x^{2}+x^{3}+x^{4}+x^{5}=\frac{1-x^{6}}{1-x}$;

$\triangleright$ for the potential student visiting a church, it is the same: $1+x=\frac{1-x^{2}}{1-x}$.

Thus, the global generating series is:

$$
\frac{1}{1-x^{2}} \frac{1}{1-x} \frac{1}{1-x^{6}} \frac{1-x^{6}}{1-x} \frac{1-x^{2}}{1-x}=\frac{1}{(1-x)^{3}}
$$

By expanding this last expression into a generating series (already done in class), we find that the coefficient of $x^{83}$ is $\frac{84 \cdot 85}{2}=3570$.