Time complexity of variation on Coupon's collector problem

I need to know the complexity of the following algorithm:

Draw a set of n numbers from a larger set of m numbers, one by one, randomly, with replacement. The result may be any set of numbers as long as the size is n and the elements are different.

This is a variation of the Coupon collector's problem: we do not have to draw all m elements, any set of different numbers of size n will do.

Example:

n = 3, m = 5, result = {1,5,2} produced by successive draws 1,1,5,1,5,2 from pool {1,2,3,4,5}

What is the average time complexity of this problem?

Answer

The expected
number of trials until your collection has $n$ distinct elements is:

$${m\over m}+{m\over m-1}+{m\over m-2}+\cdots +{m\over m-n+1}.$$