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Turkish NMO First Round - 2012 Problem - 32 {Combinatorics}

Source:

July 1, 2012

Problem Statement

How many permutations (a1,a2,,a10)(a_1,a_2,\dots,a_{10}) of 1,2,3,4,5,6,7,8,9,101,2,3,4,5,6,7,8,9,10 satisfy a11+a22++a1010=4|a_1-1|+|a_2-2|+\dots+|a_{10}-10|=4 ?
<spanclass=latexbold>(A)</span> 60<spanclass=latexbold>(B)</span> 52<spanclass=latexbold>(C)</span> 50<spanclass=latexbold>(D)</span> 44<spanclass=latexbold>(E)</span> 36 <span class='latex-bold'>(A)</span>\ 60 \qquad <span class='latex-bold'>(B)</span>\ 52 \qquad <span class='latex-bold'>(C)</span>\ 50 \qquad <span class='latex-bold'>(D)</span>\ 44 \qquad <span class='latex-bold'>(E)</span>\ 36