ID Numbering
The Federal Bureau of Investigation (FBI) maintains a records system that stores civil background checks and criminal histories in a database. Each file has its own identification number (ID). With the initial system for generating IDs, the system would generate only 400 million unique IDs. Because this was not sufficient for the number of individuals in the database, more numbers were needed. The new algorithm uses eight characters, where each character can be a digit from 0 to 9 or one of 17 letters of the alphabet (letters that can be confused with numbers, like the number 1 and letter I, are excluded).
b. The FBI does not allow the first character to be 0. To the nearest billion, how many identification numbers are possible?

Verified step by step guidance

1. Identify the total number of possible characters for each position in the ID. Each character can be a digit (0-9) or one of 17 letters, so the total number of possible characters per position is \$10 + 17 = 27$.
2. Since the ID has 8 characters, if there were no restrictions, the total number of possible IDs would be \$27^8$.
3. However, the first character cannot be 0. Since digits include 0-9, the first character can be any of the 27 characters except 0, so the number of options for the first character is \$27 - 1 = 26$.
4. For the remaining 7 characters, there are no restrictions, so each can be any of the 27 possible characters.
5. Calculate the total number of possible IDs by multiplying the number of options for the first character by the number of options for the remaining characters: \$26 \times 27^7$. This gives the total number of valid IDs under the given restriction.

Key Concepts

Counting Principle (Multiplication Rule)

The counting principle states that if there are multiple stages in a process, the total number of outcomes is the product of the number of choices at each stage. For example, if each character in an ID can be chosen independently, the total number of IDs is the product of the number of options for each character.

Permutations with Restrictions

When generating sequences like ID numbers, restrictions on certain positions (e.g., the first character cannot be 0) reduce the number of possible outcomes. This requires adjusting the count for that position while keeping the counts for other positions unchanged.

Base Counting Systems and Character Sets

The total number of possible IDs depends on the size of the character set for each position. Here, each character can be a digit (0-9) or one of 17 letters, excluding confusing characters, making a total of 27 possible characters per position, except where restrictions apply.