Expectation of multinomial distribution Three fair dice are cast. In 10 independent casts, let X be the number of times all three faces are alike and let Y be the number of times only two faces are alike. Find the joint pdf of X and Y and compute E(6XY).

You can find the joint probability mass function of a multinomial distribution.
Use that and the definition of expectation:
$$\mathsf E(6XY) = \sum_{x=0}^{10}\sum_{y=0}^{10-x} 6xy \;\mathsf P(X=x, Y=y)$$
