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# help counting

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In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color, and at lest three of the squares are red.  How many ways are there to color the five squares?

Nov 2, 2022

#1
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In a row of five squares, each square is to be colored either red, yellow, or blue, so that

no two consecutive squares have the same color, and at lest three of the squares are red.

How many ways are there to color the five squares?

Under the rules, reds can't be next to each other, so the three reds have to go in squares 1, 3, and 5.

Square 2 can have either yellow or blue.  Each of those choices leaves only one color left for square 4.

So there are only 2 ways to color the five squares.

1st way         Red Yellow Red Blue Red

2nd way        Red Blue Red Yellow Red

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Nov 2, 2022
#2
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OOPS, too late to edit my above.

I didn't read the problem properly.

I didn't account for same color in 2 and 4.

There are 4 ways.

Red  Yellow  Red  Blue  Red

Red  Blue  Red  Yellow  Red

Red  Yellow  Red  Yellow  Red

Red  Blue  Red  Blue  Red

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Guest Nov 2, 2022
#3
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I’ll interpret your question as follows. You have an  n×n  grid of  n2  cells. You’ve got one color. Each square can be either colored or left uncolored. (I’ll use 0 for uncolored and 1 for colored.) How many ways can the cells be colored so that no two rows have the same number of colored cells and no two columns have the same number of colored cells. www.mywawavisit.com

Nov 3, 2022
edited by Joehollywood  Nov 3, 2022
edited by Joehollywood  Nov 3, 2022