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f(X)$ be the number of ways of dividing the line of n boys and n girls

Source: 1972 Hungary - Kürschák Competition p2

October 15, 2022
combinatorics

Problem Statement

A class has n>1n > 1 boys and nn girls. For each arrangement XX of the class in a line let f(X)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)=0f(X) = 0 be AA, and the number of arrangements with f(X)=1f(X) = 1 be BB. Show that B=2AB = 2A.
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