The owner of a clothing factory is a math enthusiast and organizes his products into packages whose number of items is a prime number. For example, the green-colored pieces, he organizes into packages of 2 pieces each; the blue-colored pieces, into packages of 3 pieces each; the pink-colored pieces, into packages of 5 pieces each; the white-colored pieces, into packages of 7 pieces each; and so on. In a large box, the factory owner decides to place only packages of green, blue, and pink colors (it can have all three colors or just one, including only one color). The box must contain exactly 20 pieces. In how many different ways can he organize this box?

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Solution

The colors green, blue, and pink have 2, 3, and 5 pieces per pack, respectively. We can take various quantities of packs of each color, including not taking any, as long as the total number of pieces is exactly 20. Let's create a table to list the possibilities:

| Green | Blue | Pink |
| :---: | :---: | :---: |
| 10 | 0 | 0 |
| 7 | 2 | 0 |
| 6 | 1 | 1 |
| 5 | 0 | 2 |
| 4 | 4 | 0 |
| 3 | 3 | 1 |


| Green | Blue | Pink |
| :---: | :---: | :---: |
| 2 | 2 | 2 |
| 1 | 6 | 0 |
| 1 | 1 | 3 |
| 0 | 5 | 1 |
| 0 | 0 | 4 |

Therefore, there are 11 different ways.