First, It appears that you are mixing up the definition of conditional probability for events and for random variables.
The definition $P(a,b,c)=P(c|a,b)P(a,b)=P(c|a,b)P(b|a)P(a)$ is for events.
For random variables, the definition is :
$$P(A = a,B = b,C =c)=P(C = c|A = a,B = b)P(B = b,A = a)$$
where $a,b,c$ is the values that the random variables $A$, $B$, $C$ can assume (in this case, $0$ or $1$).
Now, if you want the conditional probability of $C$, given $A$ and $B$, you must obtain the probability $P(B = b,A = a)$.
Here is what you should do:
First, obtain $P(B = b, A = a)$ for all possible values of $a$ and $b$. For example, to obtain $P(B = 0, A = 1)$, just sum all probabilities where $B = 0$ and $A = 1$. Do the same for the other possible values of $A$ and $B$.
- Now, you can calculate $P(C = c|A = a,B = b) = \frac{P(A = a,B = b,C =c)}{P(B = b,A = a)}$
you can follow the same reasoning for the other conditional probabilities.
A good exercise for learning is to obtain all possible combinations of conditional and unconditional probabilities for all variables.