+136 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?
\(\boxed{\phantom{X}} < \boxed{\phantom{X}} > \boxed{\phantom{X}} < \boxed{\phantom{X}} > \boxed{\phantom{X}}\)
+1145 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.
