Find the probability that the ball ends up in the slot in the middle

Solution 1:

Question: How did the author of the solution get C(4,2)? I mean what does the 4 and the 2 represent here?

This means choosing 2 out of 4. See https://en.wikipedia.org/wiki/Combination

So exactly as you said, choosing 2 to be turning left out of 4 possible turns. The rest are right turns.