2. Cinderella, together with her fairy godmother, released a collection of crystal slippers featuring 7 new models. The fairy tale heroines organized a presentation of the collection for some guests: the audience was supposed to say which slippers they liked. The guests wrote in a questionnaire which models they thought were the best. It is known that no two guests chose the same set of favorite slippers. What is the maximum number of people (excluding Cinderella and the fairy) who could have attended the presentation?
Solution. Suppose that the guests were given a questionnaire where they could put a "1" next to each pair of shoes they liked and a "0" next to the pair they did not select as the best. Thus, the opinion of each guest can be recorded as a string of "1"s and "0"s. Then, the number of different sets of favorite shoes is the same as the number of evaluation strings, which means the maximum number of guests at the presentation is $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2=2^{7}=128$ people.

## Answer: 128 guests.