The 62 interns from the Animath internship each take one ice cream ball from 5 flavors offered. We know that

- the malabar-flavored ice cream was at least as popular as the tabasco-flavored ice cream;
- the number of students who chose cactus-flavored ice cream is a multiple of 6;
- at most 5 students tasted lemon basil ice cream;
- at most one student opted for foie gras ice cream.

When Igor places the order with the ice cream vendor, he communicates, in alphabetical order of the flavors, the number of students who chose each flavor. How many different lists of numbers could he have pronounced?
It is a matter of counting the number of ways to write 62 in the form $r+t+6a+b+c$ where $r, t, a, b, c$ are natural numbers, $r \leqslant t, b \leqslant 5$ and $c \leqslant 1$. Since we want $r \leqslant t$, we can write $t=r+s$ with $s \in \mathbb{N}$. It is therefore sufficient to count the number of ways to write 62 in the form $2r+s+6a+b+c$ with $r, s, a, b, c$ natural numbers, $b \leqslant 5$ and $c \leqslant 1$. This corresponds to the coefficient of degree 62 of the generating series

$$
\left(\sum_{r \geqslant 0} x^{2r}\right)\left(\sum_{s \geqslant 0} x^{s}\right)\left(\sum_{a \geqslant 0} x^{6a}\right)\left(1+x+\ldots+x^{5}\right)(1+x)
$$

Using the closed form of the geometric series, this series can be rewritten as

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

This last rational fraction is the derivative of $\frac{1}{2} \frac{1}{(1-x)^{2}}$, so its series expansion is $\sum_{n \geqslant 2} \frac{n(n-1)}{2} x^{n-2}$, obtained by differentiating $\frac{1}{2} \sum_{n \geqslant 1} n x^{n-1}$. Consequently, the coefficient of degree 62 we are looking for is $\frac{64 \times 63}{2}=2016$.