Bookstore Promotional Code Capacity
An independent bookstore is launching a rewards program and will print one 4-character promotional code on each receipt. Codes use only uppercase English letters (A–Z) and digits (0–9). To reduce typing errors, the first character must be a letter, and the remaining three characters may be letters or digits. Every code must contain at least one digit somewhere after the first character. Repetition of characters is allowed. The owner wants to know whether the scheme can supply 900,000 unique codes, because the chain projects up to that many receipts in the first year. A part-time employee proposes a simpler scheme: three letters followed by two digits, also allowing repetition, because 'it looks more professional.' The owner also asks how many of the original 4-character codes contain exactly one digit, since those may be easier for customers to read aloud.
1. Count the total number of valid 4-character codes under the original rules. Show how you handle the first character and the at-least-one-digit condition.
Read the worked answer
First character: 26 choices (letters only). Each of the last three positions: 36 choices (26 letters + 10 digits), so without the digit condition there are 26 × 36³ = 26 × 46,656 = 1,213,056 codes. Now subtract codes with no digit after the first character. Those last three positions are letters only: 26 choices each, so 26³ = 17,576. Thus at least one digit gives 26 × (36³ - 26³) = 26 × (46,656 - 17,576) = 26 × 29,080 = 756,080 valid codes.
2. How many of the original valid codes contain exactly one digit after the first character? Explain the choice of positions.
Read the worked answer
Choose which one of the last three positions holds the digit: 3 choices. Put a digit there: 10 choices. The other two of the last three positions must be letters: 26² choices. The first character is a letter: 26 choices. Total = 3 × 10 × 26² × 26 = 3 × 10 × 676 × 26 = 527,280. This counts exactly one digit because the other two non-first positions are forced to be letters, so no second digit can appear.
3. Does the original scheme meet the 900,000-code requirement? Compare it with the employee's three-letters-plus-two-digits scheme, and recommend a practical option.
Read the worked answer
The original scheme gives 756,080 codes, which is 900,000 - 756,080 = 143,920 short. It cannot supply 900,000 unique codes. The employee's scheme has 26³ × 10² = 17,576 × 100 = 1,757,600 codes, more than enough. But it changes the format to five characters and fixes exactly two digits, so it is not simply a cosmetic improvement; it changes the user experience and the counting rules. The 'more professional' reason is aesthetic, not a capacity argument. A practical alternative is to keep four characters but allow the first character to be a digit as well, or to add a fifth character. Either change must be checked against readability and the 900,000 target, not chosen only because it looks better.