Three $6$-sided fair dice are rolled. In $10$ independent throws, Let $A$ be the number of times all the sides are the same and let $B$ be the number of times only two sides are the same. Find $E(6AB)$:
Here is how I approached this:
I know that $E(6AB) = 6E(AB)$. I know that $E(AB)$ for a multinomial distribution is:
$n(n-1)\cdot\text{probability of }A\text{ occurring}\cdot\text{probability of }B\text{ occurring}$
Probability of $A$ occurring is $6\left(\dfrac16\right)^3$. Probability of $B$ occurring is $6\left(\dfrac16\right)^2 \left(\dfrac56\right)$
Therefore $E(AB) = 10 \cdot 9 \cdot 6 \left(\frac16\right)^3 \cdot 6 \left(\frac16\right)^2 \cdot \frac56$ and $E(6AB)$ is just $6$ times the quantity to the left.
I would like to know if my approach above is correct since I don't get the correct solution of $25/4$ Thanks for the help.