RO-TST 2014 Problem 18

Let $m$ be a positive integer and let $A$ , respectively $B$ , be two alphabets with $m$ , respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$ . Let $a_n$ be the number of words of length $n$ , formed with letters from $A$ , in which appear all the letters from $A$ , each an even number of times. Let $b_n$ be the number of words of length $n$ , formed with letters from $B$ , in which appear all the letters from $B$ , each an odd number of times. Compute $\frac{b_n}{a_n}$ .