The number of positive integral solutions of the inequality x + y + z ≤ 20 is

Step-by-step solution

The number of solutions (positive integer) of x + y + z = r is r - 1 C 2 ∴ The desired number of solutions is ∑ r = 3 20   r - 1 C 2 = 2 C 2 + 3 C 2 + 4 C 2 + . . . . . . . . + 19 C 2 = 3 C 3 + 3 C 2 + 4 C 2 + . . . . . . . . + 19 C 2 = 4 C 3 + 4 C 2 + . . . . . . . . + 18 C 2 + 19 C 2 = 20 C 3 (Adding in the same way) = 20 × 19 × 18 3 3 × 2 = 114 0