2014 Fermat Problem 24

Mohammed has eight boxes numbered 1 to 8 and eight balls numbered 1 to 8. In how many ways can he put the balls in the boxes so that there is one ball in each box, ball 1 is not in box 1, ball 2 is not in box 2, and ball 3 is not in box 3?

\textbf{(A)}\ 27\,240\quad \textbf{(B)}\ 29\,160\quad \textbf{(C)}\ 27\,360\quad \textbf{(D)}\ 27\,600\quad \textbf{(E)}\ 25\,200

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