Its easy to see that (y+z) is even. Hence let (y+z) = 2t.
The given equation is then x+t=10 which has 9 solutions one for each value of t, .
It easy to see that for each t, we have to solve y+z = 2t, which will have 2t-1 solutions.
Hence the total number of solutions = 1+3+5+...+17 = 92 = 81.