TITLE: In a random selection of three pairs among $6$ people what is the probability that each girl will be matched with her boyfriend?
QUESTION [5 upvotes]: There are $6$ people, $3$ boys and $3$ girls. Each boy is in a relationship with one girl. Three pairs are randomly drawn. What is the probability that these three pairs will be the actual couples?

My reasoning was
$$
P =  \frac{3}{\binom{6}{2}} \times \frac{2}{\binom{4}{2}} \times \frac{1}{\binom{2}{2}}$$
But this does not give me the answer a professor has given me. I'd appreciate some hints or new ways of approaching the problem.

REPLY [4 votes]: Line up the girls.
There are $3! = 6$ ways of pairing the boys with the girls, only one of which is correct.
Thus $Pr = \dfrac16$