Proof that Conditional of Poisson distribution is Binomial
HINT
Remember that $$\sum_{i=0}^{n} \binom{n}{i} \lambda^i \mu^{n-i} = \left( \lambda + \mu\right)^n$$ The above gives us that $$\sum_{i=0}^{n} \dfrac{n!}{i! (n-i)!} \lambda^i \mu^{n-i} = \left( \lambda + \mu\right)^n$$ which inturn gives us that $$\sum_{i=0}^{n} \dfrac{ \lambda^i \mu^{n-i}}{i! (n-i)!} = \dfrac{\left( \lambda + \mu\right)^n}{n!}$$