Consider the total number of binary words of length n (2ⁿ) and think about how many of them contain at least one '1' (since the shortest contained string of 1's must be of length 1).

Recognize that the problem reduces to counting the number of binary words that contain at least one '1' and no substring of two or more consecutive '1's (to ensure the shortest contained string of 1's is exactly length 1).

Use combinatorial reasoning to derive the count: subtract the number of binary words with no '1's (all '0's) and the number of binary words with at least one substring of two or more consecutive '1's from the total number of binary words (2ⁿ).