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?
\(\[\boxed{\phantom{X}} < \boxed{\phantom{X}} > \boxed{\phantom{X}} < \boxed{\phantom{X}} > \boxed{\phantom{X}}\]\)
There are no restrictions on the numbers in the first and last boxes, so they can be filled in in 5 ways each. However, the number in the middle box must be 3, and the number in the middle box must be either 2 or 4.
If the number in the middle box is 2, then the number in the third box must be 1, and the number in the fourth box must be 4. This can be done in 3 ways.
If the number in the middle box is 4, then the number in the third box must be 3, and the number in the fourth box must be 2. This can also be done in 3 ways.
Therefore, there are a total of 5⋅5+3+3=31 ways to fill in the boxes so that all the inequalities are true.
