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Difficulty: MediumPermutations, Combinations, and Fundamental Counting Principle

A cybersecurity system generates 5-character identification codes consisting of 3 distinct letters followed by 2 distinct digits. The letters must be selected from the set {A,B,C,D,E,F}\{A, B, C, D, E, F\} and the digits from the set {1,2,3,4}\{1, 2, 3, 4\}. If the first character of the code must be a vowel (AA or EE), how many such distinct 5-character identification codes can be formed?

  1. A
    120
  2. B
    240
  3. 480Answer
  4. D
    720
  5. E
    1,440

Answer

480 distinct 5-character identification codes can be formed.
To construct the 5-character code, break the process into sequential choices for each position: Position 1 must be a vowel (AA or EE), giving 2 options. Position 2 can be any of the remaining 5 distinct letters. Position 3 can be any of the remaining 4 distinct letters. Position 4 (the first digit) can be any of the 4 available digits. Position 5 (the second digit) can be any of the remaining 3 distinct digits. Multiplying these independent choices together yields 2×5×4×4×3=4802 \times 5 \times 4 \times 4 \times 3 = 480.

Step-by-Step Solution

1
Calculate the number of ways to choose the first letter (must be a vowel)
There are 2 choices (AA or EE).
The problem restricts the first character of the code to a vowel.
2
Calculate the number of choices for the second and third letters
5 choices for the second letter and 4 choices for the third letter.
The letters must be distinct, leaving 5 remaining letters from the original pool of 6 for the second position, and 4 for the third position.
3
Calculate the number of choices for the two distinct digits
4 choices for the first digit and 3 choices for the second digit.
There are 4 digits available in {1,2,3,4}\{1, 2, 3, 4\} and they must be distinct.
4
Apply the Fundamental Counting Principle to find total codes
2×5×4×4×3=4802 \times 5 \times 4 \times 4 \times 3 = 480.
Multiply the number of independent choices for each sequential position.

Key Concept

Fundamental Counting Principle with Permutations and Restrictions
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Topics

Permutations, Combinations, and Fundamental Counting Principle

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