The number $\text{N}$ is between $9$ and $17$.  The [average](https://artofproblemsolving.com/wiki/index.php/Average) of $6$, $10$, and $\text{N}$ could be
$\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$
We know that $9<N<17$ and we wish to bound $\frac{6+10+N}{3}=\frac{16+N}{3}$.
From what we know, we can deduce that $25<N+16<33$, and thus \[8.\overline{3}<\frac{N+16}{3}<11\]
The only answer choice that falls in this range is choice $\boxed{\text{B}}$