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How many arrangements of the numbers {1, 2, 3, ..., 7, 8} are there where the sum of any two adjacent numbers is odd?

 

\(\phantom{1, 2, 3, \dots, 7}\)

 Jun 19, 2022
 #1
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There are 1400 arrangements.

 Jun 20, 2022
 #2
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Without the repetitions of the 8 digis, you have the following permutations:

 

For the first 2 pairs, there are: 4 x 4 ==16 permutations. For the nest 2 pairs, there are: 3 x 3 ==9 permutations

 

So, the total beginning with digit will be: 16  x  9 ==144

 

144 x 8 digits ==1,152 permutations

 Jun 20, 2022

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