Half the people in a room left.  One third of those remaining started to dance.  There were then $12$ people who were not dancing.  The original number of people in the room was
$\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 42 \qquad \text{(E)}\ 72$
Let the original number of people in the room be $x$.  Half of them left, so $\frac{x}{2}$ of them are left in the room.
After that, one third of this group is dancing, so $\frac{x}{2}-\frac{1}{3}\left( \frac{x}{2}\right) =\frac{x}{3}$ people are not dancing.
This is given to be $12$, so \[\frac{x}{3}=12\Rightarrow x=36\]
$\boxed{\text{C}}$