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Turkey NMO 2000 1st Round - P23 (Combinatorics)

Source:

July 25, 2012

Problem Statement

A committee with 2020 members votes for the candidates A,B,CA,B,C by a different election system. Each member writes his ordered prefer list to the ballot (e.g. if he writes BACBAC, he prefers BB to AA and CC, and prefers AA to CC). After the ballots are counted, it is recognized that each of the six different permutations of three candidates appears in at least one ballot, and 1111 members prefer AA to BB, 1212 members prefer CC to AA, 1414 members prefer BB to CC. How many members are there such that BB is the first choice of them?
<spanclass=latexbold>(A)</span> 5<spanclass=latexbold>(B)</span> 7<spanclass=latexbold>(C)</span> 8<spanclass=latexbold>(D)</span> 10<spanclass=latexbold>(E)</span> More information is needed <span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ \text{More information is needed}