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=(8!)/(4!(8-4)!)= (8*7*6*5)/(4*3*2)=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:

`(12)/(70)= (6)/(35)`

The probability is
`(6)/(35)`