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Turkish NMO First Round - 2012 Problem - 04 {Combinatorics}

Source:

July 1, 2012

Problem Statement

How many f:AAf : A \rightarrow A are there satisfying f(f(a))=af(f(a)) = a for every aA={1,2,3,4,5,6,7}a \in A=\{1,2,3,4,5,6,7\}?
<spanclass=latexbold>(A)</span> 1<spanclass=latexbold>(B)</span> 106<spanclass=latexbold>(C)</span> 127<spanclass=latexbold>(D)</span> 232<spanclass=latexbold>(E)</span> None <span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 106 \qquad <span class='latex-bold'>(C)</span>\ 127 \qquad <span class='latex-bold'>(D)</span>\ 232 \qquad <span class='latex-bold'>(E)</span>\ \text{None}