Question 13
(IND) In town $A$, there are $n$ girls and $n$ boys, and each girl knows each boy. In town $B$, there are $n$ girls $g_1,g_2,\ldots,g_n$ and $2n-1$ boys $b_1,b_2,\ldots,b_{2n-1}$. The girl $g_i$, $i=1,2,\ldots,n$, knows the boys $b_1,b_2,\ldots,b_{2i-1}$, and no others. For all $r=1,2,\ldots,n$, 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 $A(r)=B(r)$ for each $r=1,2,\ldots,n$.