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how many sequences x_1,x_2,x_3,...,x_7 can be formed in which all the x_i are integers greater than 0 and less than 6, and no two adjacent are equal

 Dec 10, 2021
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Without loss of generality, assume that the first number is 1. Then, x_2 can be any of 2, 3, 4, or 5 (assume it is 2, WLOG again), which is 4 possibilities. x_3 can thus be 1, 3, 4, or 5. It is apparent that the pattern of 4 possibilities continues throughout x_4, x_5, x_6, and x_7, so we have 5 possibilities for the first number, then 4 possibilities for the six following numbers, totaling to \(5 \cdot 4^6 = 20480.\)

 Dec 21, 2021

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