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Difficulty: HardFundamental Counting Principle

An airline creates a 55-character identification code for each of its cargo routes. The characters in the code must satisfy the following sequential rules from left to right:

1. The first character must be a non-zero even digit.
2. The second and third characters must be distinct uppercase vowels chosen from the set {A,E,I,O,U}\{A, E, I, O, U\}.
3. The fourth character must be a single-digit prime number.
4. The fifth character must be a single digit (00 through 99) such that the sum of the first digit and the fifth digit is odd.

How many such unique identification codes can the airline create?

Answer: 1600 codes

Answer

1,600 unique identification codes
To find the total number of unique identification codes, we evaluate the number of available options for each character position and multiply them together using the Fundamental Counting Principle:

1. First position (non-zero even digit): The even digits from 0 to 9 are 0,2,4,6,80, 2, 4, 6, 8. Excluding 0 leaves 44 options (2,4,6,82, 4, 6, 8).
2. Second and third positions (distinct vowels): There are 55 vowels (A,E,I,O,UA, E, I, O, U). The second position has 55 options, and the third position has 44 remaining options, yielding 5×4=205 \times 4 = 20 combinations.
3. Fourth position (prime digit): The single-digit primes are 2,3,5,72, 3, 5, 7, giving 44 options.
4. Fifth position (parity condition): The sum of the first digit and fifth digit must be odd. Since the first digit is an even number, the fifth digit must be an odd number to make the sum odd. The single-digit odd numbers are 1,3,5,7,91, 3, 5, 7, 9, giving 55 options.

Multiplying the choices for all positions gives:
4×20×4×5=1,6004 \times 20 \times 4 \times 5 = 1,600

Step-by-Step Solution

1
Determine the number of valid choices for the 1st position (non-zero even digit).
4 choices (2,4,6,82, 4, 6, 8).
The digit 0 is excluded because the first character must be non-zero.
2
Determine the number of valid choices for the 2nd and 3rd positions (distinct vowels).
5×4=205 \times 4 = 20 choices.
There are 5 vowels available for the 2nd position, and 4 remaining vowels for the 3rd position since they must be distinct.
3
Determine the number of valid choices for the 4th position (prime digit).
4 choices (2,3,5,72, 3, 5, 7).
The single-digit prime numbers are 2, 3, 5, and 7.
4
Determine the number of valid choices for the 5th position (parity constraint).
5 choices (1,3,5,7,91, 3, 5, 7, 9).
The sum of an even integer and another integer is odd if and only if the second integer is odd. Since the 1st digit is guaranteed to be even, the 5th digit must be odd.
5
Apply the Fundamental Counting Principle across all 5 sequential stages.
4×20×4×5=1,6004 \times 20 \times 4 \times 5 = 1,600 unique codes.
Each stage represents an independent decision stage with a fixed number of valid choices.

Key Concept

Fundamental Counting Principle with Restricted Positions and Parity Rules
Estimated Time:2m 0s
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Topics

Fundamental Counting Principle

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