Probability of Choosing Exactly 1 pair out of 4

Question

Gretchen has eight socks, two of each color: magenta, cyan, black, and white. She randomly draws four socks. What is the probability that she has exactly one pair of socks with the same color?

Here is my thinking. First, we need to find the total amount of outcomes. I thought that the denominator would be $\binom{8}{4}$. However, there are multiple socks of the same color. This would throw my reasoning off. However, I know that the numerator must be $4\cdot3\cdot2= 24$. Help is greatly appreciated.

Answer 1

There are $4$ possible pairs Gretchen can pick, and $\binom62 - 3 = 12$ ways for her to pick socks of two other colors. There are $\binom84=70$ total ways Gretchan can pick socks, and thus there is a $\frac{4\cdot12}{70}=\frac{24}{35}$ probability Gretchen picks exactly $1$ pair.

This number might seen large, but considering there is a $\frac{2^4}{70} = \frac8{35}$ chance of picking no pairs, and a $\frac{\binom42}{70}=\frac3{35}$ chance of two pairs, it adds up (to $1$).

Answer 2

About finding the numerator,you can think with the same way as in a card's deck : we have 52 cards and we categorise them on 13 numbers where every number contains 4 signs. In this case we have 4 colours and every colour contains 2 socks. So in order to get exactly one pair of socks you have to: get one colour from four(you can do it in 4 ways),and from that colour take both socks(there is only one way). Then,from the other three colours pick 2 colours(you can do that in 3 ways) and from each colour pick one sock(you can do it in 2 ways).By multiplying the numbers 4*3*2 you get the numerator you ask about.