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# Combinatoric Question

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A bag contains red and blue tiles. Each tile has a number from the set written on it. I want to arrange of these tiles in a row, so that the numbers on any three consecutive tiles sum to . In how many ways can this be done, assuming that there are an unlimited number of tiles for any color and number combination?

Aug 8, 2019

#1
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I think you forgot to type in what the numbers sum to.

Aug 8, 2019
#2
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$$\text{Suppose you want the numbers of 3 consecutive tiles to sum to N}\\ \text{Further let's assume that the set you refer to has M numbers in it}\\ \text{We want the number of 3 element partitions of N from the elements of the set}\\ \text{This is equivalent to sorting N balls into M bins}\\ \text{The number of ways this can be done is given by the stars and bars problem (google) and is}\\ \dbinom{N+M-1}{M-1}\\ \text{Tiles can additionally be colored red or blue. This gives an additional factor of 2^L,\\ where L is the length of the row of tiles.}\\ \text{So if your row is length L the number of arrangements is }\\ 2^L \dbinom{N+M-1}{M-1}$$

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Aug 8, 2019
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