If things are chosen randomly from n possible choices, then number of selections that all possible choices are selected at least once is approximated by $nH_n$, where $H_n$ is $nth$ Harmonic Number.
The lower bound of number of selections seems to be $n$ i.e in a rare event, one may be able to select all choices at least once after n selections.
Is this possible? is this lower bound right?
What is the likelihood of such a rare event?
What is the probability that a collection of size x has all the possible choices at least once
What is the probability P(x, n) such that it takes x or less than x draws to obtain all n distinct elements, when drawing with replacement from a set of n distinct elements.
I ran a simulation 10000 times for n=100 ($nH_n$=521), it shows that minimum_number_of_selections = 279 and maximum_number_of_selections = 1302.
Yes, $n$ is possible to achieve if all the $n$ first draws are different.
The probability is $\frac{n!}{n^n} = \frac{(n-1)!}{n^{n-1}}$ because you need to get a permutation of the objects but all in all you can get any sequence.