NP Complete Clique Variation
You can prove this by a reduction from the Hitting Set problem (which is NP-Complete):


  Given a base set $U$, a family $F = \{S_1,\ldots,S_\ell\}$ of subsets of $U$ and an integer $q$, decide if there exists a set $H$ of size at most $q$ which intersects every set in $F$ (this is called a hitting set).


So suppose given an instance of the Hitting Set problem. For every element in $u\in U$, create a new TA $T_u$. For every set $S\in F$, create a new class $C_S$ whose TAs correspond to the elements in $S$. Let $k = |U|-q$. 

Then if you interpret the TAs being left out of the room as the elements chosen in $H$ it is not hard to see that there exists a hitting set of size at most $q$ if and only if you can put $k$ TAs in the room. That is because putting $k = |U|-q$ TAs in the room is equivalent to leaving at most $q$ outside of the room. And hitting every set in $F$ is equivalent to having at least one TA per class left out of the room.

(I leave the formal proof up to you)