how do you ban things from subsets How can you get the total number of subsets, if you generate all subsets from the set $\{A,B,C,D,E,F\}$ but BAN all subsets that either include $A$ and $B$, $B$ and $C$, or $C$ and $D$. 

Is there a way to do this non-manually?

I would also appreciate links and general information about the topic!
There are $2^6$ total subsets.  Now let us subtract the banned ones-there are $2^4$ subsets containing $A$ and $B$ and also $2^4$ with each other pair.  We have subtracted the $2^3$ that have $A,B,C$ twice and the $2^3$ that have $B,C,D$ twice, so we have to add them back in once.  Finally the $2^2$ that have $A,B,C,D$ have been counted, subtracted three times, and added twice, so no more correction is necessary.  The final answer is 
$$2^6-3\cdot 2^4+2\cdot 2^3=32$$
This is an example of the inclusion-exclusion principle.
To "ban" $A$ and $B$, take all the subsets of:
$$
\{C,D,E,F\}.
$$
In more detail, first carry out the calculation
$$
\{A,B,C,D,E,F\} \setminus \{A, B\},
$$
which results in the set $\{C,D,E,F\}$.  Take all subsets of this set.