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TOT 220 1989 Spring S4 a club of 11 people has a committee

Source:

March 7, 2021
combinatorics

Problem Statement

A club of 1111 people has a committee. At every meeting of the committee a new committee is formed which differs by 11 person from its predecessor (either one new member is included or one member is removed). The committee must always have at least three members and , according to the club rules, the committee membership at any stage must differ from its membership at every previous stage. Is it possible that after some time all possible compositions of the committee will have already occurred?
(S. Fomin , Leningrad)