Number of integral solutions to the equation x + y + z = 21 , where...

We have,

$x \ge 1, y \ge 3, z \ge 4$

$⇒ x - 1 \ge 0, y - 3 \ge 0, z - 4 \ge 0$

So,

$x + y + z = 21$

$⇒ \left(x - 1\right) + \left(y - 3\right) + \left(z - 4\right) = 13$

So, number of non-negative integral solutions is

$= C3 - 1 13 + 3 - 1 = C2 15 = \dfrac{15 \times 14}{2} = 105$