KURSCHAK 1968 Problem 3

For each arrangement $X$ of $n$ white and $n$ black balls in a row, let $f(X)$ be the number of times the color changes as one moves from one end of the row to the other. For each $k$ such that $0 < k < n$ , show that the number of arrangements $X$ with $f(X) = n -k$ is the same as the number with $f(X) = n + k$ .