Let \(S\) be the set of all passwords which are six to eight characters long, where each character is either an alphabet from \(\{ A , B , C , D , E \}\) or a number from \(\{1,2,3,4,5\}\) with the repetition of characters allowed. If the number of passwords in \(S\) whose at least one character is a number from \(\{1,2,3,4,5\}\) is \(\alpha \times 5^{6}\), then \(\alpha\) is equal to \(.......\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.