A stadium has 24 rows of 10 seats each. Describe at least two ways mrs Allen's class of 36 students can sit if an equal number of students sits in each row.

Question

A stadium has 24 rows of 10 seats each. Describe at least two ways mrs Allen's class of 36 students can sit if an equal number of students sits in each row.

Answer 1

To answer this question Vicki, we have to see how the class can sit in the rows. With the restriction of equal number in each row, we have these possibilities: use 18 rows two each, 12 rows three each, 9 rows four each, and 6 rows six each. So that is four arrangements, but each has many options for choice of row and position in the row. Each particular arrangement can have 36! possibilities for the students to sit. One of the challenges of such a problem is how to express the answer. So I'll get you started on working out the details, and leave you to finish it. You can always ask for more help. Let's define a "sitting" as anytime the class sits down in their row patterns. Each sitting has 36! ways to occur. Let's deal with the first of the four arrangements noted above, the use of 18 rows with two each. How many ways can that occur? To pick rows, you have 24!/(24-18)! or 24!/6!. In each row chosen in that count, you have two seats (not necessarily together) so 10!/(10-2)! or 10*9 or 900 ways for two students to sit in one row. The sitting is then: which rows, which seats, which students.  So Vicki, using this approach you can build up the total number, expressed as several terms each involving factorials. Thank Mrs. Allen for the idea!