The total number of 3 - digit numbers whose sum of digits is 10 , is ..........

Step-by-step solution

Let x y z be the three-digit number such that x + y + z = 10 ,   x ≥ 1 ,   y ≥ 0 ,   z ≥ 0 Let x - 1 = t ⇒ x = 1 + t So, x ≥ 1 ⇒ t + 1 ≥ 1 ⇒ t ≥ 0 Also, x + y + z = 10 ⇒ t + y + z = 10 - 1 = 9 So, t + y + z = 9 ,   0 ≤ t ,   y ,   z ≤ 9 Thus, total number of non-negative integral solution = C 3 - 1 9 + 3 - 1 = C 2 11 = 11 . 10 2 = 55 But for t = 9 ,   x = 10 , so required number of integers = 55 - 1 = 54 .