So this problem was asked in my final term exam and I couldn't solve it, I ended up finding a lot of terms like P(A,C) and P(B,C) but I couldn't come up with a way to have all the three A,B and C in a single term.
How would we solve this.
EDIT : The question said find the probability if it's possible to do so, so I guess the answer was we couldn't.
Either the question was wrong or you copied it wrong.
Consider an experiment where you flip two coins $a, b$. Let $A$ be the event "coin $a$ was heads", and $B$ be "coin $b$ was heads". Now let $C_0$ be "the coin flip results were different", and let $C_1$ be "the coin flip results were the same".
Then for $C \in \{C_0, C_1\}$ you have that $$P(C \mid A) = P(A) = P(C \mid B) = P(B) = 1/2.$$ However, for $C_0$ you have $P(C_0 \mid A, B) = 0$, while for $C_1$ you have $P(C_1 \mid A, B) = 1$. Thus you cannot find $P(C \mid A, B)$ with only the values in your title.