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In how many ways can the numbers 1 through 5 be entered once each into the five boxes below so that all the given inequalities are true?

 

\(\boxed{\phantom{X}}>\boxed{\phantom{X}}>\boxed{\phantom{X}}<\boxed{\phantom{X}}<\boxed{\phantom{X}}\)

 Nov 13, 2022
 #1
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The number of ways is 44.

 Nov 13, 2022
 #2
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In how many ways can the numbers 1 through 5 be entered once each into the five boxes below so that all the given inequalities are true? 

 

–––  >  –––  >  –––  <  –––  <  –––   

 

Firset, consider the middle box.  

 

It can be only a 1, because that's the only number in the list that there are four others larger than it. 

 

So, for the middle box, there can be only a 1.  After that, the rest is easy. 

 

                                 5 > 4 > 1 < 2 < 3   

                                 5 > 3 > 1 < 2 < 4   

                                 3 > 2 > 1 < 4 < 5   

                                 4 > 2 > 1 < 3 < 5   

 

These are FOUR ways to do it.  Maybe you can think of some others that I missed.  But there aren't 44, that's for sure.  

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 Nov 13, 2022
edited by Guest  Nov 13, 2022
edited by Guest  Nov 13, 2022
edited by Guest  Nov 13, 2022
edited by Guest  Nov 13, 2022
edited by Guest  Nov 13, 2022

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