Find the number of integral solutions of the equation:
$a+b+c=m$ with $0\gt a\gt b\gt c$
And the generalized version:
$a_1 + a_2 + \cdots + a_k = m$ with $ 0\gt a_1\gt a_2\gt \cdots \gt a_k$
2026-03-27 18:26:42.1774636002
Number of solutions of the two equations
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This is the number of partitions of a given integer into a sum. Say, $$m = 6,$$ and we would like to write it as the sum of 3 positive summands. To do this, write $$ 6 = 1 + 1 + 1 + 1 + 1 + 1. $$ A sum is formed by choosing 2 of the 5 pluses in the above expression; e.g, $$ 1 + 1 +' 1 + 1 + 1 +' 1 = 2 + 3 + 1. $$ The number of ways to choose 2 pluses is "5 choose 2".