Mean squared distance of a ten-step random walk

Leaving a party, Marc steps out of the door and starts wandering along a straight street. Every minute he takes a single step, to the left or to the right with probability $\tfrac12$ each, independently of his earlier steps. Write $D_n$ for his displacement from the door (in steps) after $n$ minutes. What is $\mathbb{E}[D_{10}^2]$?

$10$. Write $D_n$ as a sum of $n$ independent $\pm1$ steps. Squaring, every step contributes $1$ and every cross term has mean $0$ because the steps are independent with mean zero, so $\mathbb{E}[D_n^2]=n$; here that is $10$.