# Probability Of 4 Decks of cards

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?

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

• Note: The larger number should be the prescript. $^{100}C_{47}$ Commented Apr 14, 2016 at 1:30
• Why do you think you are selecting 47 from 100 cards? There are 104 red cards and 104 black cards. Also you have not taken account of ways to select 53 black cards to go with those 47 red to make 100. This should be out of all the ways to select any 100 cards from all 208. Commented Apr 14, 2016 at 1:33
• Why do you have deleted this question ? math.stackexchange.com/questions/1777258/… Commented May 8, 2016 at 21:17

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 $\binom{104}{47}\binom{104}{53}$.
Remark: By $\binom{n}{k}$ I mean the number of ways to choose $k$ objects from $n$ objects. This is $\frac{n!}{k!(n-k)!}$.