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Given 5 colors to choose from, how many ways can we color the four unit squares of a 2×2 board, given that two colorings are considered the same if one is a rotation of the other? A color can be used more than once

Mathgenius Dec 14, 2018

#1**0 **

We have 5 options and must choose 4. However, the same color can be used more than once. 5*4 = 20, and \({20}\choose{4}\) = 4845.

We did overcount (Example: 1234 is the same as 2341, but we counted it twice), so to take away overcounting, we divide 4845 by 5 to get 969. With some other casework and steps shown below, you will end up with \(\boxed{165}\).

Do not try to list out the answers because there are many and that is inefficient. Always use number theory.

- PM

PartialMathematician Dec 15, 2018