In town A, there are n girls and n boys, and each girl knows each boy. In town B, there are n girls g1,g2,…,gn and 2n \minus{} 1 boys b_1, b_2, \ldots, b_{2n\minus{}1}. The girl gi, i \equal{} 1, 2, \ldots, n, knows the boys b_1, b_2, \ldots, b_{2i\minus{}1}, and no others. For all r \equal{} 1, 2, \ldots, n, denote by A(r),B(r) the number of different ways in which r girls from town A, respectively town B, can dance with r boys from their own town, forming r pairs, each girl with a boy she knows. Prove that A(r) \equal{} B(r) for each r \equal{} 1, 2, \ldots, n. combinatoricscountingCombinatorial Identitybijectionbinomial coefficientsIMO Shortlist