A customer ordered fifteen Zingers that are to be placed in packages of four, three or one. In how many ways can this order be fulfilled?

### Solution to Zinger packaging box puzzle

#### What is known:

- More than one box of a specific size to be used in every combination.
- Any number of specific size package may be used in a combination.
- No restriction on not using any specific size box in a combination.

The puzzle boils down to finding **all possible ways a sum of 15 can be achieved by using additions of numbers of 4, 3 and 1.**

#### A systematic method to find all possible combinations of packaging

**Strategy:**

- Find all possible combinations using 4.
- Find all possible combinations using 3 and 1 excluding 4 (already found).
- Find all possible combinations using 1, excluding 4 and 3 (already found).

**Packages using 4 Zinger box: in each product, first term is the number of boxes and second term the size of box:**

- 3x4 + 1x3 = 15: maximum possible three 4 Zinger boxes with maximum one 3 Zinger box,
- 3x4 + 3x1 = 15: three four Zinger boxes, zero 3 Zinger box and three one Zinger box,
- 2x4 + 2x3 + 1x1 = 15, number of 4 Zinger boxes reduced by one, maximum two 3 Zinger boxes used ,
- 2x4 + 1x3 + 4x1 = 15, with same two 4 Zinger boxes, number of 3 Zinger boxes reduced by 1,
- 2x4 + 7x1 = 15, zero 3 Zinger box used,
- 1x4 + 3x3 + 2x1 = 15, number of 4 Zinger boxes reduced further by 1 to the minimum of 1,
- 1x4 + 2x3 + 5x1 = 15,
- 1x4 + 1x3 + 8x1 = 15,
- 1x4 + 11x1 = 15, no 3 Zinger box used.

These are all possible combinations using 4 Zinger box. Total number 9.

**Packages using 3 Zinger box and 1 Zinger box:**

- 5x3 = 15, maximum five 3 Zinger boxes used,
- 4x3 + 3x1= 15, number of 3 Zinger boxes reduced by 1, same with next combinations,
- 3x3 + 6x1 = 15,
- 2x3 + 9x1 = 15,
- 1x3 + 12x1 = 15.

Total 5. Cumulative 14.

**Packages using only 1 Zinger box:**

- 15x1 = 15, maximum fifteen 1 Zinger boxes needed.

Total 1.

**Cumulative 15** possible package combinations. These are all possible.

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