I need to solve the following problem: we generate m random numbers where each: $\forall n, n\in [2,20] $

find the probability function of variable $X$ where $X= max(n_1 , n_2 ...,n_m )$

Now, I think I managed to solve it using the following "hack": $$ p(x\leq k)=(\frac{k-1}{19})^m $$ $$p(x=k)=p(x\leq k) - p(x\leq k-1)$$

But when I try to solve it the straight forward way (using inclusion exclusion) I can't simplify the equation:

$$ p(x=k)=\sum_i^m\left(\begin{array}{c}19\\ i\end{array}\right)\cdot (\frac{1}{19})^i\cdot (\frac{k-1}{19})^{m-1}\cdot (1)^{m-1} $$

this looks like Newton's binomial, but I cant solve it because the -1

Any help will be appropriated


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