# Counting The Sixes

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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