A closet contains n pairs of shoes. If 2r shoes are chosen at random, (where 2r < n), what is the probability that the chosen shoes will contain no matching pair?
I have tried thinking about this problem on my own and have not gotten much farther than deducing that there are ${2n \choose 2r}$ possible combinations that could be chosen.
When I did some searching I found two different solutions for this same problem and I am having trouble discerning which would be correct. No explanation is given with either solution and I am relatively new to probability and combinations so any help from someone with a bit more experience would be much appreciated.
Solution 1:
The probability is computed as $ {n \choose 2r}2^{2r}/{2n \choose 2r} $
Solution 2:
The probability is computed as $ {n \choose r}2^{r}/{2n \choose 2r} $
If anyone could tell me which of these two is correct and why. I understand that the probability is given as a certain number of combinations out of the total number of possible combinations. What do the two different factors in the numerators represent, and how do they differ in the two solutions?