Harmonic numbers probability similar to coupon collector We're ordering beer with uniform probability and with replacement. I calculated the expected value of receiving $n$ different brands of beer from some company is $E(X) = n\cdot H_n$. I've defined a random variable as:
$X = $ total number of distinct brand received. 
With every brand we receive, it slightly decreases our chance of receiving an unique brand of beer. 
If someone places $m$ amount of orders for beer, the what is the expected value? What is the number of distinct amounts of brands of beer will we receive if we only order m amounts where there exists n amount of brands 
I'm puzzled on how to compose a general equation to solve this.

Your question is less than clear, but I suppose you have $n$ distinct and equally probably brands and you take a sample of size $m$ with replacement, receiving $X$ distinct brands.
You can work out the probablity of not getting a particular brand as $\left(1-\frac1n\right)^m$ and so the expected number of distinct brands received is $$E[X]=n\left(1- \left(1-\frac1n\right)^m\right)$$ which for large $n$ is close to $n \left(1- e^{-m/n}\right)$.
