Number of integral solutions to the equation $$x + y + z = 21$$, where $$x \geq 1, y \geq 3, z \geq 4$$, is equal to ______.

Find the number of integral solutions to $$x + y + z = 21$$ where $$x \geq 1$$, $$y \geq 3$$, $$z \geq 4$$.

Substitute to remove lower bounds.

Let $$a = x - 1 \geq 0$$, $$b = y - 3 \geq 0$$, $$c = z - 4 \geq 0$$.

Then $$x = a + 1$$, $$y = b + 3$$, $$z = c + 4$$, and:

$$(a + 1) + (b + 3) + (c + 4) = 21$$

$$a + b + c = 13$$

Apply stars and bars.

The number of non-negative integer solutions to $$a + b + c = 13$$ is:

$$\binom{13 + 2}{2} = \binom{15}{2} = \frac{15 \times 14}{2} = 105$$

The correct answer is 105.