Define as the expected number of matching pairs of dice when there are black dice and white dice. Define as the probability that a randomly chosen white die matches at least one of the black dice. Then gives a recurrence relationship for the expected value.

since this is the probability that among the black dice, none of them rolled the same as the one white die.

Hence, and

Next, I would recommend looking into generating functions (or calculate for small using a statistics engine like R).