Stars and bars with a twist

Question

Consider $5$ labeled bins and $200$ unlabeled balls. We have the standard stars and bars formula $\binom{200 + 5 - 1}{5 - 1}$ ways to place them in the bins with no restriction.

What if $x_1 \ge x_2 \ge x_3 \ge x_4 \ge x_5 \gt 0$?

Answer 1

If $x_1 \ge x_2 \ge x_3 \ge x_4 \ge x_5 \gt 0$, then each $x_i$ is at least $1$. Consider any integer partition of size $5$ on $200$. For any partition, say $$50 + 100 + 25 + 20 + 5 = 200$$ There's always a non-increasing arrangement of the partitions. For example, the above can be arranged to look like $$100 + 50 + 25 + 20 + 5 = 200$$ Hence, $$x_1 = 100, x_2 = 50. x_3 = 25, x_4 = 20, x_5 = 5$$ So the question is an integer partition of size $5$, i.e. $P_5(200)$