Choosing sides

Angel chooses a face. Lets call this face A.

In order for none of the faces to be adjacent, Bob has five faces to choose from, namely the five that are neither opposite or adjacent to the one that Angel chose. Lets call this face B. There is a $\dfrac{5}{11}$ chance he will pick a valid face.

Note : He can't have chosen the one opposite to A since then all the remaining faces are adjecent to either A or B.

Finally, C has two faces to choose from. Namely, the two of the five B had to choose from that aren't adjacent to B (or B itself!) There are $\dfrac{2}{10}$ ways to do this.

So, the probability that none of the three faces are adjacent is given by:

$P = \dfrac{5}{11} \cdot \dfrac{2}{10} = \dfrac{1}{11}$

$1+11 = \boxed{12}$