Eighteen people are standing in a socially-distanced line to enter a grocery store. Five people are wearing a black mask, $6$ are wearing a gray mask, and $7$ are wearing a white mask. Suppose that these $18$ people got in line in a random order. The expected number of pairs of adjacent people wearing different-colored masks can be given by $\frac{a}{b}$, where $\gcd(a, b) = 1$. Compute $a+b$.

Let's think step by step and output the final answer within \boxed{}.
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