Expected Payoff of a Three N-Sided Dice Matching Game

You roll three fair $N$-sided dice, with faces numbered 1 through $N$. If all three dice show the same number, you receive $A$ dollars. If exactly two of them show the same number, you receive $B$ dollars. If all three numbers are different, you receive $C$ dollars; in another version of the question you pay $C$ dollars instead. What is the expected payoff of a single roll?

Give the answer as an expression in $N$, $A$, $B$ and $C$ for both versions, then evaluate it for the values you are given. The specific numbers are generated randomly, and each question states the required rounding.

Constraints and Clarifications

• $N$ is a positive integer, and the three dice are fair and independent.
• $A$, $B$ and $C$ are nonnegative amounts. The version of the question decides whether the all-different outcome pays $C$ or costs $C$.
• There is a single roll and no separate entry fee unless the question states one.

Clarifying Questions

• Does "exactly two the same" exclude the case where all three match?
• In the version where the all-different outcome costs money, is $C$ the whole loss, or is an entry fee charged as well?
• Should the answer be rounded only at the end, and to how many decimal places?