We have a standard deck of 52 cards, that has 26 red and 26 black cards.
We draw n cards from the deck at the same time, then return them and reshuffle. We repeat that k times.
The question: What is the probability that we drew each red card at least once, and we never draw any black card?
I tried to use: $\binom{52}{26} P(\textrm{exactly all red cards are drawn at least once}) = P(\textrm{we draw exactly 26 different cards})$.
Then $P(\textrm{we draw exactly 26 different cards}) = P(\textrm{we draw exactly 26 different cards | we draw 1st card at least once}) P(\textrm{we draw 1st card at least once}) + P(\textrm{we draw exactly 26 different cards | we never draw 1st card}) P(\textrm{we never draw 1st card}) = P(\textrm{we draw exactly 25 different cards from a pile of 51}) (1 - (\binom{51}{n} / \binom{52}{n})^k) + P(\textrm{we draw exactly 26 different cards from a pile of 51}) (\binom{51}{n} / \binom{52}{n})^k .$
Then I wanted to repeat it, but it got too complicated, so I was wondering if there is an easier way.