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Given 5 colors to choose from, how many ways can we color the four unit squares of a 2\times 2 board, given that two colorings are considered the same if one is a rotation or reflection of the other? (Note that we can use the same color for more than one square.)

 Aug 19, 2025
 #2
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All the same color = 5 ways

2 the same color and 2 different = 5C3*3*3 = 90ways

3 the same and one different = 5C2*2 = 20 ways

2 of one color and 2 of other = 5C2*2 =20ways

all colors different = 5C4*3*2= 30 ways

5+90+20+20+30 = 165 ways

Credits to melody

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 Aug 19, 2025
edited by bqimath  Aug 19, 2025

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