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A pair of 12-sided fair dice with faces numbered $1,2,3, \ldots \ldots, 12$ is rolled. The probability that the sum of the numbers appearing has remainder 2 when divided by 9 is
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The correct answer is:
$\frac{1}{9}$
\begin{array}{l|c}
\mathrm{x}_{1}+\mathrm{x}_{2}=11 or & \mathrm{x}_{1}+\mathrm{x}_{2}=20 \\
possible cases & \\
\qquad(1,10) & (8,12) \\
(2,9) & (9,11) \\
(3,8) & (10,10) \\
(4,7) & \\
(5,6) & \\
\end{array}
$\frac{10}{144}$ $+\frac{6}{144}=\frac{16}{144}=\frac{1}{9}$
\mathrm{x}_{1}+\mathrm{x}_{2}=11 or & \mathrm{x}_{1}+\mathrm{x}_{2}=20 \\
possible cases & \\
\qquad(1,10) & (8,12) \\
(2,9) & (9,11) \\
(3,8) & (10,10) \\
(4,7) & \\
(5,6) & \\
\end{array}
$\frac{10}{144}$ $+\frac{6}{144}=\frac{16}{144}=\frac{1}{9}$
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