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Find the number of ways of arranging the numbers 1,2 ,3, 4, 5, 6 in a row so that the product of any two adjacent numbers is at least 5.

 Feb 20, 2024
 #1
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At least 5 means we need 1 in between 5 and 6. imagine this as a block. So including the block there are 4 total numbers. These numbers have \(4!=24\) ways to arrange them. Then the block has 2 permutations. There can be anyother number next to each other as it is always greater than 5 so:

\(24\times2=\boxed{48}\)

 Feb 20, 2024
 #2
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Any permutation starting 1, 5 , ...    4! of them 

Any permuttion starting 1, 6, ...       4! of them

Any permutation ending ..., 5, 1      4! of them

Any permutation ending ..., 6, 1      4! of them

Any permutation starting 5, 1, 6, ...  3! of them

Any permutation starting 6, 1, 5, ...  3! of them

etc. ...

 Feb 20, 2024

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