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?

Answer 1

Begin by computing the number of all the possibilities:
It is given by the following formula:$C_8^4=\frac{8!}{4!(8-4)!}= \frac{8\times7\times6\times5}{4\times3\times2}=70$
Computing the number of favorable cases:
Exactly one pair of socks with the same color means either one magenta pair, 
or one cyan pair, or one black pair or one white pair:
Number of possibilities with one magenta pair means"a magenta pair plus two socks with different colors from three possibilities" means 3 possibilities. 
Same computation for the other colors. 
Adding we get: $3+3+3+3=12\text{ possibilities }$
By definition, the probability of an event is the number of favorable cases over the number of possible cases:
$\frac{12}{70}= \frac{6}{35}$

The probability is $\frac{6}{35}$