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

Let $xyz$ be the three-digit number such that

$x + y + z = 10, x \ge 1, y \ge 0, z \ge 0$

Let $x - 1 = t ⇒ x = 1 + t$

So, $x \ge 1 ⇒ t + 1 \ge 1 ⇒ t \ge 0$

Also, $x + y + z = 10 ⇒ t + y + z = 10 - 1 = 9$

So, $t + y + z = 9, 0 \le t, y, z \le 9$

Thus, total number of non-negative integral solution $= C3 - 1 9 + 3 - 1 = C2 11 = \dfrac{11 . 10}{2} = 55$

But for $t = 9, x = 10$, so required number of integers $= 55 - 1 = 54$.