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.