In the cafeteria, there are six dishes to choose from. Each day, Vasya takes some set of dishes (possibly not taking any dishes), and this set of dishes must be different from all the sets he took on previous days. What is the maximum number of days Vasya can follow such rules, and on average, how many dishes will he eat per day?

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Each set of dishes can be matched with an opposite set, consisting exactly of those dishes that are not in the original set.

## Solution

The number of days is obviously the number of different sets of six dishes. For each dish, there are two possibilities - to be chosen or not chosen. Therefore, the number of days is $2^{6}$.

Each set of dishes can be matched with an opposite set, consisting exactly of those dishes that are not in the original set. Together, the original and the opposite set contain six dishes, meaning on average there are three dishes per set. Since all 64 sets are divided into pairs of opposite sets, on average Vasya ate three dishes over these 64 days.

## Answer

64 days, on average 3 dishes per day.