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?
Begin by computing the number of all the possibilities: It is given by the following formula:[tex]C_8^4=\frac{8!}{4!(8-4)!}= \frac{8\times7\times6\times5}{4\times3\times2}=70 [/tex] 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: [tex]3+3+3+3=12\text{ possibilities }[/tex] By definition, the probability of an event is the number of favorable cases over the number of possible cases: [tex]\frac{12}{70}= \frac{6}{35} [/tex]
