troubleshooting Question

# How many different colors in sample

Math / Science
Dear experts,

first of all I should point out, that this is not homework. I rephrased a problem I have at work in classical analogies of probability theory.

Suppose you have an urn with balls of different colors. There are n colors and k balls of each color. You draw m (m << n) balls. Lets say with replacement, but it is not really important.

Question: What is the probability that the sample contains balls of exactly c different colors (not matter which ones).

Given:
n = number of colors
k = number of balls for each color
m = number of balls drawn
c = number of different colors in the sample

Probability of c given n,k and m = P(c | n,k,m)

I am really looking for a closed form,like a formula/method which takes n,k,m and c.

I am not interested in specific examples like

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