The students having eaten a bit too much ice cream, Igor goes to the market to buy fruits for the 62 trainees of the Animath training. How many baskets of 62 different fruits can he assemble knowing that

- apples are sold in lots of 2;
- bananas are sold in lots of 5;
- there are only 4 oranges left
- there is only one pear left.

## - Partitions -

A partition of a strictly positive integer $n$ is a representation of $n$ as a sum of other strictly positive integers, considered up to permutation of the terms. For example, 4 can be partitioned as $1+1+1+1, 1+1+2, 2+2, 1+3, 4$. Generating series constitute a very powerful tool for dealing with problems on partitions.
We will solve the problem more generally for $n$ interns. It is about determining the number of ways to write $n$ in the form $n=2a+3b+c+d$ with $a, b, c, d$ being natural numbers and $c \leqslant 4, d \leqslant 1$, so we can write $x^n$ as $x^{2a} x^{3b} x^c x^d$, which is also the coefficient of $x^n$ in the expansion of:

$$
\underbrace{\left(\sum_{k \geqslant 0} x^{2k}\right)}_{\text{apples}} \underbrace{\left(\sum_{k \geqslant 0} x^{5k}\right)}_{\text{bananas}} \underbrace{\left(1+x+x^2+x^3+x^4\right)}_{\text{oranges}} \underbrace{(1+x)}_{\text{pears}}.
$$

This generating function is

$$
\frac{1}{1-x^2} \frac{1}{1-x^5} \frac{1-x^5}{1-x}(1+x)=\frac{1}{(1-x)^2}=2x+3x^2+4x^3+\ldots
$$

and the coefficient of degree $n$ is therefore $n+1$. In particular, the solution to the problem is 63.