Maximum of Multiple Dice Rolls
You roll $k \geq 1$ fair $n$-sided dice, each showing a value in $\{1, 2, \ldots, n\}$. Let $M_k$ denote the maximum value among all $k$ rolls.
Find a closed-form expression for $P(M_k = r)$, the probability that the maximum equals $r$, for any
\leq r \leq n$.
Then compute the answer explicitly for $k = 3$, $r = 4$, and $n = 6$.
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