Problem Statement
Suppose you have a fair dice with $$$m$$$ faces, numbered from $$$1$$$ to $$$m$$$. You roll this dice $$$n$$$ times independently, and each face has an equal probability of $$$ \frac{1}{m} $$$. Find the expected value of the maximum number obtained after these $$$n$$$ rolls.

Expected Maximum
If you roll an $$$m$$$-faced dice $$$n$$$ times, then there are $$$m^n$$$ possible combinations of rolls. The expected maximum is the average of the maximum values obtained from all $$$m^n$$$ combinations of rolls.

Intuition:
We have to compute the expected maximum value. That is