Number of integral solutions to the equation x + y + z = 21 , where x ≥ 1 , y ≥ 3 , z ≥ 4 , is equal to _____ .

Step-by-step solution

We have, x ≥ 1 ,   y ≥ 3 ,   z ≥ 4 ⇒ x - 1 ≥ 0 ,   y - 3 ≥ 0 ,   z - 4 ≥ 0 So, x + y + z = 21 ⇒ x - 1 + y - 3 + z - 4 = 13 So, number of non-negative integral solutions is = C 3 - 1 13 + 3 - 1 = C 2 15 = 15 × 14 2 = 105