How many ordered triples of pairwise distinct, positive integers are there such that ?
Clarification: Ordered triples - and count as different solutions.
Clarification: Pairwise distinct - a set of integers are pairwise distinct if no two of them are the same. For example, the set is NOT pairwise distinct, hence does not count as a solution.
Set . We first count the number of triples without restriction. We seek the number of non-negative integer solutions to . This is equivalent to arranging 6 1's with 2 0's (dividing blocks), thus there are ways to solve either of the two equations. By the rule of product, there are solutions in total.
If , we seek the number of non-negative integer solutions to . For each equation, there are 4 solutions, corresponding to . By the product rule, there are solutions in all. The cases and are exactly the same. If , there is exactly 1 solution.