Coin Toss Game

The Problem

A fair coin is tossed $n$ times. Let $X$ be the number of heads.

1. What is $\mathbb{P}(X=k)$ for $0\le k\le n$?
2. What is $\mathbb{E}[X]$ and $\text{Var}(X)$?
3. Game variant: You win \2^k$ifyougetexactly$k$ heads. What is the fair price of this game (the expected payoff)? Is it finite?