Alphabet theory, again.

This equation can be rewritten as $bcd + acd + abd + abc = abcd$ . Now since $a,b,c,d$ are all odd, each of $bcd, acd, abd$ and $abc$ is odd, and the sum of 4 odd numbers is even. But $abcd$ is odd, so the given equation cannot be satisfied with odd positive integers. The desired answer is then $\boxed{0}$ .