Conditional Distribution of a Binomial Sum Component

Let $X \sim \text{Binomial}(m, p)$ and $Y \sim \text{Binomial}(n, p)$ be independent.

(a) What is the distribution of $S = X + Y$? State and justify.

(b) Derive the conditional PMF $P(X = k \mid S = s)$ for valid $k$.

(c) Identify this conditional distribution by name and parameters. Interpret the result: why does $p$ disappear from the conditional distribution?

(d) Verify with a numerical example: $m = 10$, $n = 15$, $p = 0.4$. Compute $P(X = 3 \mid S = 8)$.