How do I use the pigeonhole principle for these questions?
A drawer contains $6$ pairs of black, $5$ pairs of white, $5$ pairs of red, and $4$ pairs of green socks.
(a) How many single socks do we have to take out to make sure that we take out two socks with the same color. (pigeonhole)
(b) How many single socks do we have to take out to make sure that we take out two socks with different colors? (not really pigeonhole)
For a, the maximum number of different color socks is four, so if we pull five we must have a pair. The four colors are the pigeonholes.
For b, you could take out all twelve black socks first, so we have to pull thirteen to make sure we get two different.