Length of a conference

Task number: 3456

Professor Baldy has found out that 5 of his friends are attending the same conference as he. During the lectures at the conference, of these 5 people he encounters:
– every individual \(10\times\),
– every pair \(5\times\),
– every triple \(3\times\),
– every quadruple \(2\times\),
– all five \(1\times\).

How many lectures were at the conference, given that the professor encountered at least one of his friends at each lecture?

  • Solution

    The entire group of five must have been present at one lecture. At that lecture, the professor of course encountered every group of 4, 3, 2 and 1.

    We must account for the remaining quadruples, for which we must have 5 lectures, during which each triple is encountered \(2\times\), each pair \(3\times\) and each individual \(4\times\).

    The triples are now all accounted for.

    To account for the remaining pairs we need 10 lectures, in which each individual is encountered \(4\times\) more.

    To account for the remaining individuals we need 5 more lectures.

  • Answer

    The conference had 21 lectures.

Difficulty level: Moderate task
Reasoning task
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