For a positive integer $$n$$, a distinct 3-partition of $$n$$ is a triple $$(a, b, c)$$ of positive integers such that $$a < b < c$$ and $$a+b+c = n$$. For example, $$(1, 2, 4)$$ is a distinct 3-partition of 7. The number of distinct 3-partitions of 15 is
Solution
Fix $$a$$ and count the pairs $$b < c$$ with $$b + c = 15 - a$$ and $$b > a$$. For $$a = 1$$ the pairs are $$(2,12), (3,11), (4,10), (5,9), (6,8)$$, for $$a = 2$$ there are 4, for $$a = 3$$ there are 2 and for $$a = 4$$ there is 1, while $$a \ge 5$$ gives none. The total is $$5+4+2+1 = 12$$.
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