Exercise 10. A hospital has just received a shipment of vaccines that need to be stored at very low temperatures in special refrigerators. They received 300 vials. Knowing that each of the 5 special refrigerators in the hospital can hold more than 300 vials (the vaccination campaign has just begun!) and that all the vials are identical and interchangeable, in how many ways can the vials be distributed among the refrigerators?
Solution to Exercise 10 Since the flasks are interchangeable, the problem reduces to counting the number of 5-tuples $\left(a_{1}, \ldots, a_{5}\right)$ of positive integers whose sum is 300 (the number $a_{i}$ corresponding to the number of flasks in fridge number i).

Choosing a 5-tuple $\left(a_{1}, \ldots, a_{5}\right)$ is equivalent to choosing the values of the sums $a_{1}, a_{1}+a_{2}, a_{1}+a_{2}+a_{3}, a_{1}+a_{2}+a_{3}+a_{4}$. Indeed, the choice of these sums determines the value of each integer, noting that $a_{5}=300-\left(a_{1}+a_{2}+a_{3}+a_{4}\right)$.

If we set $b_{i}=\sum_{k=1}^{i} a_{k}$, we need to calculate the number of ways to choose a 4-tuple $\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ of integers satisfying $0 \leqslant b_{1} \leqslant b_{2} \leqslant b_{3} \leqslant b_{4} \leqslant 300$.

To count this, consider a sequence of 304 aligned points. We then choose 4 to replace with lines. We associate with each line the number of points to its left. We verify that the sequence of numbers obtained is indeed a non-decreasing sequence of length 4 with values in $\{0, \ldots, 300\}$.

![](https://cdn.mathpix.com/cropped/2024_06_04_f521a17d739d2f1da72bg-16.jpg?height=217&width=572&top_left_y=1255&top_left_x=820)

Choosing a non-decreasing sequence of 4 integers from $\{0, \ldots, 300\}$ is equivalent to choosing 4 points out of 305 to replace with lines, without regard to order. Therefore, there are a total of $\binom{304}{4}$ ways to proceed.

Grader's Comment: The exercise is well solved.