# In a random selection of three pairs among $6$ people what is the probability that each girl will be matched with her boyfriend?

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.

• When the first pair is “randomly drawn,” is it any of the $6\choose2$ pairs of two people, or is it any of the $3\times3$ pairs of a boy and a girl? You and the professor have interpreted this differently. – Steve Kass Mar 4 '16 at 14:21
• The question explicitly states "Each boy is in a relationship with one girl" – true blue anil Mar 4 '16 at 17:19

There are $3! = 6$ ways of pairing the boys with the girls, only one of which is correct.
Thus $Pr = \dfrac16$