Jolana is preparing refreshments for her friends - open-faced sandwiches. She spreads them with potato salad and wants to add more toppings: ham, hard cheese, a slice of egg, and a strip of pickled pepper. However, she does not want any two of her sandwiches to have the exact same combination of toppings. What is the maximum number of different sandwiches can she create, given that none of them should have all four toppings and none should have only the salad (i.e., no additional toppings)?

(M. Petrová)

Idea. Come up with a suitable system for listing the individual options.
For clarity, we will create a table. An asterisk means that the given open-faced sandwich contains the respective ingredient, while an empty cell means that the sandwich does not contain that ingredient.

|  | ham | cheese | egg | pepper |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1 topping | $*$ |  |  |  | 1 |
|  |  | $*$ |  |  | 2 |
|  |  |  | $*$ |  | 3 |
| 2 toppings | $*$ | $*$ |  | $*$ | 4 |
|  | $*$ |  | $*$ |  | 5 |
|  | $*$ |  |  | $*$ | 7 |
|  |  | $*$ | $*$ |  | 8 |
|  |  | $*$ |  | $*$ | 9 |
| 3 toppings | $*$ | $*$ | $*$ | $*$ | 10 |
|  | $*$ | $*$ |  | $*$ | 12 |
|  | $*$ |  | $*$ | $*$ | 13 |
|  |  | $*$ | $*$ | $*$ | 14 |

Since we created the table systematically and exhausted all possibilities, we can see that Jolana can prepare up to 14 open-faced sandwiches that meet her requirements.