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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 19, 2022
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We know that 5 is the greatest number, so we'll take two cases:

(Also note that since 1 is the smallest number, it will have to go in the middle box, no number is less than 1 in the list)

 

Case 1. 5 is on the leftmost side of this inequality.

Then we will get:

5 > __  > 1 < __ < __

 

    Subcase 1.1: 4 is on the rightmost side:

    5 > __ > 1 < __ <  4
    Now 2 or 3 can be placed in each of those 2 boxes, so resulting in 2 for this subcase.

 

   Subcase 1.2: 4 is next to 5:

   5 > 4 > 1 < __ <__

   Note that there is only 1 option for this subcase as we can only place 2 and 3 in only one way. 

 

Hence there are 3 options for this case. 

 

 

Case 2. 5 is on the rightmost side of this inequality:

This inequality "mirrors" itself so this will also result in 3 options as well.

 

Hence, there will be \(\boxed{6}\) ways to order the integers 1 through 5 in the boxes. 

 Nov 19, 2022

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