State the inclusion-exclusion principle.

Let $A=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a string of $n$ digits, where $a_{i} \in\{0,1, \ldots, 9\}$. We say that the string $A$ has a run of length $k$ if there is some $j \leqslant n-k+1$ such that either $a_{j+i} \equiv a_{j}+i(\bmod 10)$ for all $0 \leqslant i<k$ or $a_{j+i} \equiv a_{j}-i(\bmod 10)$ for all $0 \leqslant i<k$. For example, the strings

$$(\underline{0,1,2}, 8,4,9),(3, \underline{9,8,7}, 4,8) \text { and }(3, \underline{1,0,9}, 4,5)$$

all have runs of length 3 (underlined), but no run in $(3,1,2,1,1,2)$ has length $>2$. How many strings of length 6 have a run of length $\geqslant 3$ ?