2001 Gauss 8 Problem 23

The integers 2, 2, 5, 5, 8, and 9 are written on six cards, as shown. Any number of the six cards is chosen, and the sum of the integers on these cards is determined. Note that the integers 1 and 30 cannot be obtained as sums in this way. How many of the integers from 1 to 31 cannot be obtained as sums?

\begin{array}{|c|c|c|}\hline 2 & 2 & 5 \\ \hline 5 & 8 & 9 \\ \hline \end{array}

\textbf{(A)}\ 4\quad \textbf{(B)}\ 22\quad \textbf{(C)}\ 8\quad \textbf{(D)}\ 10\quad \textbf{(E)}\ 6

If there are no answer choices shown, enter a numerical answer.

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