The number of integer solutions to an equation

Question

How many integer solutions are there to the equation $x_1 + x_2 + x_3 = 0$ if $x_i \geq -5$ for all $i$?

Answer: I know the answer is $C(15 + 3 - 1, 15)$, which is derived from the formula $C(r + n - 1, r)$. I understand such problems if we are using "non-negative" solutions. However, including negatives seem to be confusion and the textbook I have doesn't explain it.

Any suggestions?

Answer 1

$x_1 + x_2 + x_3 = 0$

$ \implies (x_1 + 5) + (x_2 + 5) + (x_3 + 5) = 15$

Or, $y_1 + y_2 + y_3 = 15$ where $y_i = x_i + 5 \geq 0, $ as $x_i \geq - 5, $ for $i = 1, 2, 3$.

So the number of solutions to $x_1 + x_2 + x_3 = 0$ with $x_i \geq - 5 ~$ is same as the number of non-negative integer solutions to $y_1 + y_2 + y_3 = 15$.