How many possible values of (p + q + r) are there satisfying 1/p + 1/q + 1/r = 1, where p, q and r are natural numbers (not necessarily distinct)?

We need natural numbers p, q, r with 1/p + 1/q + 1/r = 1. The well-known solutions (unordered) are: (3,3,3) giving sum 9; (2,4,4) giving sum 10; (2,3,6) giving sum 11. These are the only three solutions in positive integers. So there are exactly 3 distinct possible values of (p+q+r): 9, 10, and 11.