I'm doing random permutations of a set (shuffling) and I want to generate the minimum number of permutations to have every values at every places, at least once.
Say I have a set of $n$ distinct elements.
There's $n!$ possible permutations of that set.
In one permutation, there's $\frac{(n - 1)!}{n!} = \frac{1}{n}$ probability to get one value at a given place.
There's $\frac{n - 1}{n}$ probability of not having that value at that place.
So the probability of having one value at a given place in $m$ random permutations is $1 - (\frac{n - 1}{n})^m$
Now what's the probability of having every values at given place in $m$ random permutations ?
And what's the probability of having every values at every place in $m$ random permutations ?