🚀 go-pugleaf

Hi everyone!


Some of my friends and I have been discussing a question and
we came up w/ an answer and would like to verify if its correctness.

Please give any comments freely!!!

Thanks...

Question:

Assume there are m kinds of balls in an infinitely large pool
(i.e. there are infinitely many balls)
and the distributions of different kinds of balls are equal.
Find the probability of getting all kinds of balls if n balls
are picked.

Solution:

First group the n balls you picked into m groups according to
their kinds using m-1 sticks.  So the number of possible
arrangements for the sticks and balls is (n+m-1)C(m-1).

Then we are to find the number of possible arrangements of:
(m balls of different kinds) + (n-m balls of whatever kinds) + (m-1 sticks)
since the m balls are fixed (to satisfy the constraint that there
exists at least one ball of each kind) and what we are trying to
find is the number of arrangements of n-m balls and m-1 sticks.
So, the possible arrangements is:    ((n-m)+(m-1))C(m-1) = (n-1)C(m-1)

Therefore, the probability is:    (n-1)C(m-1)
				---------------
                                 (n+m-1)C(m-1)