A question about the law of total probability

[lola19991]

Dan put n different kinds of cheese in a row randomly, so that between two kinds of cheese there is a space. Then he puts a pickle on one of the n-1 spaces between the cheeses randomly. What is the probability of goat cheese and parmesan cheese (2 from n kinds of cheese) to be in the different sides of the pickle?

I know that I need to use law of total probability and define Ai as the pickle is found in the space number I from n-1 spaces, but I don't know how to use the formula. I would really like for a solution. The final answer is: (n+1)/(3n-3)