jmc

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 $\boxed{10}$