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P20 [Combinatorics] - Turkish NMO 1st Round - 2005

Source:

November 8, 2013
permutation statisticsInversions in permutations

Problem Statement

We are swapping two different digits of a number in each step. If we start with the number 1234512345, which of the following cannot be got after an even number of steps?
<spanclass=latexbold>(A)</span> 13425<spanclass=latexbold>(B)</span> 21435<spanclass=latexbold>(C)</span> 35142<spanclass=latexbold>(D)</span> 43125<spanclass=latexbold>(E)</span> 53124 <span class='latex-bold'>(A)</span>\ 13425 \qquad<span class='latex-bold'>(B)</span>\ 21435 \qquad<span class='latex-bold'>(C)</span>\ 35142 \qquad<span class='latex-bold'>(D)</span>\ 43125 \qquad<span class='latex-bold'>(E)</span>\ 53124