The number of triplets (x, y, z) of non-negative integers satisfying the inequality x + y + z 15 , such that x, y , and z are all distinct, is

Step-by-step solution

Let us introduce a dummy variable w 0 to convert the given inequality into an equation: x + y + z + w = 15 The total number of non-negative integer solutions to this equation is given by ¹⁵⁺⁴⁻¹C_ 4-1 = ¹⁸C₃ = 816 . Now, we subtract the cases where at least two of x, y, z are equal. Case 1: x = y The equation becomes 2x + z + w = 15 . For x = 0 , z + w = 15 16 solutions. For x = 1 , z + w = 13 14 solutions. For x = 2 , z + w = 11 12 solutions. For x = 3 , z + w = 9 10 solutions. For x = 4 , z + w = 7 8 solutions. Fo