10,11

During an interview, ten people were offered a test consisting of several questions. It is known that any five people together answered all the questions (that is, at least one of the five gave the correct answer to each question), but any four did not. What is the minimum number of questions for which this could have been the case?

#
For each question, the number of people who did not answer it is no more than four.

## Solution

According to the condition, for every group of four people, there is a question they did not answer. On the other hand, the number of people who did not answer this question is no more than four. Therefore, the number of questions is not less than the number of groups of four that can be chosen from 10 people, which is $C_{10}^{4}=210$.

Conversely, if the test had 210 questions (as many as there are groups of four people), and each question was not answered by its own group of four, then the condition of the problem is satisfied.

## Answer

With 210 questions.