Hello, Tony!

First, we'll fill the square rack.Suppose that I have 13 different spices in matching canisters and two spice racks.

The first spice rack is a square that holds 8 canisters.

It has one hole in each corner and one hole in the middle of each side for spices.

The whole square freely rotates and there are no distinguishing marks anywhere

to differentiate one side or corner from another.

The second rack is a circle with 5 holes evenly spaced around the perimeter.

Again it can spin freely and there are no distinguishing marks

to differentiate one hole from any other.

In how many different ways can I put my 13 spices into the two racks?

Choose 8 of the 13 spices.

. . There are: . choices.

If we place them in a row, there would be: . ways.

Placed in a square, there are 4 "rotations" which are identical arrangements. .**

. . Then there are: . ways.

Hence, there are: . ways to fill the square rack.

To place the remaining 5 spices in thecircularrack, there are: . ways.

Therefore, there are: . ways to fill the two racks.

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**

In our list of possible arrangements,

. . we would have, for example: .

But we'd also have: .

Since the rack rotates, these arrangements are identical.