Hello, melody!

A store has 10 identical boxed stereos to store on 3 shelves.

How many ways can the boxes be stored on the shelves if:

(a) There are no restrictions on the number of stereos on each shelf?

Place the ten boxes in a row; insert a space before, after, and between them.

. .

There are 11 spaces in which to insert 2 "dividers".

If the two dividers are placed in two different spaces,

. . there are: . ways.

For example: .

If the two dividers are placed in the same space,

. . there are: . ways.

For example: .

Hence, there are: .55 + 11 = 66 ways to place the boxes on the shelves.

Place 2 boxes on the top shelf.(b) The top shelf must have exactly 2 stereos on it.

Then the other 8 boxes are distributed to the other two shelves.

Place the 8 boxes in a row; insert a space before, after, and between them.

. .

There are 9 spaces in which to insert one divider.

Hence, there are 9 ways to place the other 8 boxes.

Therefore, there are 9 ways to have exactly two boxes on the top shelf.

For the last two parts, I had to make an exhaustive list.(c) Each shelf must have at least 2 stereos on it.

If a partition has three different numbers, for example, (1,3,6)

. . the numbers can be assigned to the shelves in ways.

If a partition has two equal numbers, for example, (3,3,4)

. . the numbers can be assinged to the shelves in ways.

. .

There are 15 ways for each shelf to have at least two boxes.

Another listing . . .(d) No shelf can have more than 5 stereos on it.

. .

There are 18 ways where each shelf has at most 5 boxes.