Partitions of 10 into exactly three parts

Text

In how many ways can $10$ be written as a sum of exactly three positive integers (order does not matter)?

Solution

Triples $a\ge b\ge c\ge 1$ with $a+b+c=10$, ordered by $c$:

• $c=1$: $a+b=9$, $b\le 4$: $\ (8,1),(7,2),(6,3),(5,4)$ → $4$
• $c=2$: $a+b=8$, $2\le b\le 4$: $\ (6,2),(5,3),(4,4)$ → $3$
• $c=3$: $a+b=7$, $b=3$: $\ (4,3)$ → $1$

Total $4+3+1=\boxed{8}$. (This is the same count as the partitions of $10$ with largest part equal to $3$: the correspondence between “exactly three parts” and “largest part $3$” comes from conjugating the diagrams.)