Consider how a "shortest contained string of 1's of length 2" can appear in an n-bit word. What patterns must exist in the word for this to occur?

Break down the problem into smaller subproblems: How many ways can a 2-length string of 1's appear in a word, and how does this relate to the placement of other bits (0's) around it?

Generalize your findings to n-bit words. How does the count scale with n, and what combinatorial approach (e.g., inclusion-exclusion, recurrence relations) can be used to derive the exact formula?