HMMT-Feb 2016 guts, Problem 9

Victor has a drawer with two red socks, two green socks, two blue socks, two magenta socks, two lavender socks, two neon socks, two mauve socks, two wisteria socks, and 2000 copper socks, for a total of 2016 socks. He repeatedly draws two socks at a time from the drawer at random, and stops if the socks are of the same color. However, Victor is red-green colorblind, so he also stops if he sees a red and green sock. What is the probability that Victor stops with two socks of the same color? Assume Victor returns both socks to the drawer at each step.

Solution: There are $\left(\begin{array}{c}2000 \\ 2\end{array}\right)+8\left(\begin{array}{l}2 \\ 2\end{array}\right)=1999008$ ways to get socks which are matching colors, and four extra ways to get a red-green pair, hence the answer. $\fbox{\frac{1999008}{1999012}}$.

Answer: \frac{1999008}{1999012}