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

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

Let $$x' = x - 1$$, $$y' = y - 3$$, and $$z' = z - 4$$, where $$x' \geq 0$$, $$y' \geq 0$$, and $$z' \geq 0$$.

Substituting into the equation:

$$(x' + 1) + (y' + 3) + (z' + 4) = 21$$

$$x' + y' + z' = 21 - 1 - 3 - 4 = 13$$

The number of non-negative integer solutions to $$x' + y' + z' = 13$$ is given by:

$$\binom{13 + 3 - 1}{3 - 1} = \binom{15}{2}$$

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

The answer is $$\textbf{105}$$.