jmc

The easiest approach is to consider how many times $6$ can appear in the units place and how many times in the tens place. If we put a $6$ in the units place, there are $10$ choices for the tens place. Likewise, if we put a $6$ in the tens place, there are $10$ choices for the units place. So, there are $\boxed{20}$ appearances of the digit $6.$

Note: Read the question carefully. There are $19$ numbers that include $6,$ but $6$ appears $20$ times. Always answer the question that is asked.