1. **Problem statement:** Find the number of runs of length $m$ in a string of length $n$. 2. **Definition:** A run is a maximal substring of consecutive identical characters. 3. **Key idea:** To count runs of length exactly $m$, consider the positions where such runs can start and ensure they are maximal. 4. **Formula:** The number of runs of length $m$ in a string of length $n$ is at most $$n - m + 1$$ because a run of length $m$ can start at any position from 1 to $n-m+1$. 5. **Explanation:** Each run of length $m$ occupies $m$ consecutive characters. To be a run, the characters before and after must be different (or boundaries of the string). 6. **Example:** For $n=10$ and $m=3$, runs of length 3 can start at positions 1 through 8. 7. **Summary:** The maximum number of runs of length $m$ in a string of length $n$ is $$n - m + 1$$, assuming the string is constructed to maximize such runs. Final answer: $$\boxed{n - m + 1}$$