KURSCHAK 1972 Problem 2

A class has $n > 1$ boys and $n$ girls. For each arrangement $X$ of the class in a line let $f(X)$ be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with $f(X) = 0$ be $A$ , and the number of arrangements with $f(X) = 1$ be $B$ . Show that $B = 2A$ .