Math Problem Statement

An oil company is going to issue new ID codes to its employees. Each code will have one letter, followed by one digit, followed by three letters. The letters W, X, Y, and Z will not be used. So, there are 22 letters and 10 digits that will be used. Assume that the letters can be repeated. How many employee ID codes can be generated?

Solution

To solve this, we need to calculate the total number of possible employee ID codes that can be generated.

Problem Analysis:

Each employee ID code follows this structure:

  1. One letter: Chosen from 22 possible letters (excluding W, X, Y, Z).
  2. One digit: Chosen from 10 possible digits (0–9).
  3. Three letters: Each chosen from the 22 possible letters, and letters can repeat.

Step-by-Step Solution:

  1. First character (letter): 22 choices.
  2. Second character (digit): 10 choices.
  3. Third character (letter): 22 choices.
  4. Fourth character (letter): 22 choices.
  5. Fifth character (letter): 22 choices.

Total Number of ID Codes:

Since each character's choice is independent of the others, multiply the number of choices for each character position:

22×10×22×22×22=224×1022 \times 10 \times 22 \times 22 \times 22 = 22^4 \times 10

Calculating this: 224=234,256and234,256×10=2,342,56022^4 = 234,256 \quad \text{and} \quad 234,256 \times 10 = 2,342,560

Answer:

The total number of employee ID codes that can be generated is 2,342,560.

Would you like further details on any step?


Here are 5 related questions for further practice:

  1. What if only letters could repeat but digits could not?
  2. How many ID codes would be possible if only specific letters (e.g., vowels) were allowed?
  3. What would be the total if digits and letters could not repeat at all?
  4. If an ID format included two digits and two letters, how many codes could be made?
  5. How would the total change if an additional restricted letter (e.g., "Q") was excluded?

Tip: When calculating permutations with repeated choices, multiply the possibilities for each slot.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Permutations

Formulas

Total combinations = Choices for each position multiplied together

Theorems

Fundamental principle of counting

Suitable Grade Level

Grades 9-12