Question: A fair coin is independently flipped $n$ times, $k$ times by $A$ and $n − k$ times by $B$. Show that the probability that $A$ and $B$ flip the same number of heads is equal to the probability that there are a total of $k$ heads.
I know the probability of getting heads or tails is the same for each because the coin is fair. I also know the probability of an arbitrary number, say, $m$ heads is equal to probability of getting $m$ tails.
So I know $P(A$ gets $x$ tails) = $P(B$ gets $x$ heads)
However, I'm confused as to where to go and how to apply this to the problem. Any help appreciated!