Ans: B
Note that the mean of the positive integers is $5$, then we have
Note that the mean of the positive integers is $5$, then we have
$\begin{array}{rcl}
\dfrac{2 + 3 + 4 + 6 + 7 + 9 + 10 + m + n}{9} & = & 5\\
m + n & = & 4
\end{array}$
Since the $9$ numbers are positive integers, therefore the possible values of $m$ and $n$ are $\left\{\begin{array}{l} m = 1 \\ n = 3\end{array}\right.$ or $\left\{\begin{array}{l} m = 2 \\ n = 2\end{array}\right.$ or $\left\{\begin{array}{l} m = 3 \\ n = 1\end{array}\right.$.
I may not be true. If $m = 1$ and $n =3$, the mode is $3$. Therefore $a = 3$.
II must be true. For the three sets of values of $m$ and $n$, the median is $4$. Therefore $b = 4$.
III may not be true. If $m = 1$ and $n =3$, the range is $9$. Therefore $c = 9$.
