Assume flower can have n possible colors equally likely, pick m flowers and there are exactly k colors.

Question

Assume $n \geqslant k$

What's the possibility? Example value $n = 12$, $m = 15$, $k = 11$

The context is that there are 12 Chinese zodiacs. Someone mentioned to me that out of 15 of his family members, they found 11 different zodiacs. Only one left. What's the odd!

I understand that all zodiacs are not equally likely, but for simplicity let's assume so.

I understand that if $n = m = k$, then $P = n!/n^n$. The first flower can pick any of the n colors, the second flower can pick from the rest n-1 colors, and so on. So it's $$\frac{n}{n}\times\frac{n−1}{n}\times\frac{n−2}{n}\times...\frac{1}{n}=\frac{n!}{n^n}$$

How about the general case? Or at least the example above?

Answer 1

EDIT : I was completely wrong. Now I edited.

Note that each of $m$ flowers has $n$ possibilities about its color.

Now, first we decide $k$ colors to choose, and second choose at least one flower in each color.

The last step is represented as $$\frac{\{(k-1)+(m-k)\}!}{(k-1)!(m-k)!}=\binom{m-1}{k-1}.$$

However, the answer is not $$\frac{\binom{n}{k}\cdot \binom{m-1}{k-1}}{n^m}.$$ This is because I also have to think about the arrangement of each set...

To be honest, I don't know how to get it.