Assume that $X\sim Binomial(n,p)$. Also, assume that $Y\sim Binomial(n+X,p)$. We are interested in computing the expected value of $Y$, i.e., $\mathop{\mathbb{E}}[Y]$.
Question: Why is the following solution incorrect?
We have: $$\mathop{\mathbb{E}}[Y] = (n+X)p$$
We take another expectation from both sides: $$\mathop{\mathbb{E}}[\mathop{\mathbb{E}}[Y]] = \mathop{\mathbb{E}}[Y] = \mathop{\mathbb{E}}[(n+X)p]=(n+\mathop{\mathbb{E}}[X])p = (n+np)p$$
The expectation is a linear form wrt to '+, const*, so $$\mathbb E(n + p X)= \mathbb E(n) + \mathbb E( p X) = n + p\mathbb E(X) = n + p^2 n $$