HMMT-Nov 2016 gen, Problem 2

I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day.

Solution: I get a matching pair on the third day with probability $\frac{1}{9}$ because there is a $\frac{1}{9}$ probability of the second sock matching the first. Given that I already removed a matching pair of the third day, I get a matching pair on the fifth day with probability $\frac{1}{7}$. We multiply these probabilities to get $\frac{1}{63}$. $\fbox{\frac{1}{63}}$.

Answer: \frac{1}{63}