A fair, $6$-sided die is rolled $20$ times, and the sequence of the rolls is recorded.

$C$ is the number of times in the 20-number sequence that a subsequence (of any length from one to six) of rolls adds up to $6.$ These subsequences don't have to be separate and can overlap each other. For example, the sequence of $20$ rolls $12334222111366141523$ contains the ten subsequences $123, 33, 42, 222, 2211, 1113, 6, 6, 141, 15$ which all add up to $6,$ so $C=10$ in this case.

The expected value of $C$ is equal to $\frac{a}{b}$ for coprime positive integers $a$ and $b.$

**What is $a+b?$**

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