Closure of Poisson under Independent Summation

Let $X \sim \text{Poisson}(\lambda)$ and $Y \sim \text{Poisson}(\mu)$ be independent.

(a) Derive the MGF of a $\text{Poisson}(\lambda)$ random variable.

(b) Using MGFs, prove that $X + Y \sim \text{Poisson}(\lambda + \mu)$.

(c) A call centre receives calls from two independent sources at rates $\lambda = 3$ and $\mu = 7$ per hour. What is the probability of receiving exactly 8 calls in the next hour?