Probability Of 4 Decks of cards

Question

I have a question about probability and I would just like to make sure im correct. This is the question:

4 standard decks, if we randomly select 100 cards without replacement find the probability of getting exactly 47 red cards. This is what I have so far.

I think its a combination because you dont care about the order of red cards being pulled out. So you would need to know how many ways to draw 47 cards out of 100. Am I supposed to put that probability over the probability of drawing 47 cards out of the total deck?

$$ P = \frac{47C_{100}}{47C_{208}} $$

Answer 1

There are $\binom{208}{100}$ equally likely ways to choose $100$ cards from $208$.

Now we count the favourables. There are $\binom{104}{47}$ ways to choose $47$ red cards from the $104$ available. For each of these ways, there are $\binom{104}{53}$ ways to choose the accompanying $53$ black cards, for a total of $\binom{104}{47}\binom{104}{53}$.

Finally, divide the number of favourables by the total number of equally likely possibilities.

Remark: By $\binom{n}{k}$ I mean the number of ways to choose $k$ objects from $n$ objects. This is $\frac{n!}{k!(n-k)!}$.