Picking three socks out of a drawer with two socks with two colors

Question

How do I show that picking 3 socks containing just black and red socks that I must get either a pair of black or red socks?

I mean it's fairly obvious, but how would I show it?

Is this pigeon hole?

Answer 1

Two holes: red and black........

Answer 2

You could show it by brute force, listing all $8$ of the possible outcomes of picking $3$ socks: BBB, BBR, BRB, RBB, BRR, RBR, RRB, and RRR. However, that’s a little messy, and the pigeonhole principle provides a neater answer. You have two pigeonholes, one labelled BLACK and the other labelled RED. You pull out $3$ socks and put each of them into the appropriate pigeonhole. Since there were more socks than pigeonholes, one pigeonhole must contain more than one sock.

Answer 3

We can use the Pigeonhole Principle (strong form). That is $q_1+q_2+⋯+q_n−n+1$ where $q_i$ is the number of desired socks of a certain color, $(2)$ and $n$ is the number of colors, $(2)$. So $2+2-2+1=3$.