First I will state the problem.
A chicken lays $n$ eggs. Each egg independently does or doesn’t hatch, with probability $p$ of hatching. For each egg that hatches, the chick does or doesn’t survive (independently of the other eggs), with probability $s$ of survival. Let $N\sim \text{Bin}(n,p)$ be the number of eggs which hatch, $X$ be the number of chicks which survive, and $Y$ be the number of chicks which hatch but don’t survive (so $X+Y =N$). Find the marginal PMF of $X$.
Intuitively, the probability that a egg hatches and chick survives is $ps$. We can consider each egg as a Bernoulli trial each with a success (hatching and surviving) probability $ps$. There are $n$ independent trials, so $X\sim \text{Bin}(n,ps)$.
But I am trying to prove this more rigorously.
For any $1\le i\le n$ we have $$P(X=i)=\sum_{j=0}^nP(X=i|N=j)P(N=j)\\=\sum_{j=i}^nP(X=i|N=j)P(N=j)\\=\sum_{j=i}^n\binom{j}{i}s^i (1-s)^{j-i}\binom{n}{j}p^j(1-p)^{n-j}$$
How do I show that the above sum collapses to $\binom{n}{i}(ps)^i(1-ps)^{n-i}$?