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