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

 Apr 20, 2024
 #1
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There are 5!=120 ways to order the numbers 1 through 5. However, this counts an arrangement as valid if it is just a rotation of another valid arrangement.

 

For instance, the arrangement 1<2>3<4>5 is the same arrangement as 5<1>2<3>4 if we just rotate the boxes cyclically.

 

There are 5 ways to rotate the boxes, so we have overcounted exactly that many times. Therefore, the number of valid arrangements is 120 ÷ 5 = 24​.

 Apr 21, 2024

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