Compute number of paths of length $2$ between two different vertices in $K_4$
My work. Since, path of length 2 would have 3 vertices,
So selecting 2 terminal vertices of a path in $\binom{4}{2}$.
And now 1 vertex needs to be selected, which can be done in 2 ways.
So, total paths must be $\binom{4}{2} \times 2=12$
But the answer is given to be 2.
What mistake I did?
I think you answered to a more difficult question. You counted ALL the paths of length $2$ between ANY two different vertices. The question is asking for the number paths of length $2$ between two given different vertices.
Call the $4$ vertices $A,B,C,D$. Given two different vertices, say $A$ and $B$, then the middle vertex of a path of length $2$ from $A$ to $B$, can be $C$ or $D$. Hence the paths are two: $A\to C\to B$ and $A\to D\to B$.