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Difficulty: MediumFundamental Counting Principle

A medical research laboratory creates a 4-digit identification code, d1d2d3d4d_1 d_2 d_3 d_4, for sample trays using digits selected from the set {1,2,3,4,5,6,7,8}\{1, 2, 3, 4, 5, 6, 7, 8\}. The digits in each code must satisfy the following conditions:
- The first digit, d1d_1, must be an odd prime number.
- The second digit, d2d_2, must be an even integer.
- All four digits in the code must be distinct.

How many different identification codes can be created under these rules?

  1. A
    18
  2. 360Answer
  3. C
    480
  4. D
    504
  5. E
    768

Answer

360
To form the 4-digit code, we calculate the number of options at each sequential stage. First, d1d_1 must be an odd prime from {1,2,3,4,5,6,7,8}\{1, 2, 3, 4, 5, 6, 7, 8\}, which gives 33 options (3,5,73, 5, 7). Second, d2d_2 must be an even integer from the set, giving 44 options (2,4,6,82, 4, 6, 8). Third, because all digits must be distinct, picking d1d_1 and d2d_2 uses 22 digits, leaving 8−2=68 - 2 = 6 choices for d3d_3 and 55 choices for d4d_4. By the Fundamental Counting Principle, the total number of codes is 3×4×6×5=3603 \times 4 \times 6 \times 5 = 360.

Step-by-Step Solution

1
Determine the number of valid choices for the first digit (d1d_1)
3 choices (3,5,73, 5, 7)
The odd prime numbers in the given set {1,2,3,4,5,6,7,8}\{1, 2, 3, 4, 5, 6, 7, 8\} are 3,5,3, 5, and 77. Note that 11 is not prime and 22 is an even prime.
2
Determine the number of valid choices for the second digit (d2d_2)
4 choices (2,4,6,82, 4, 6, 8)
The even integers in the given set are 2,4,6,2, 4, 6, and 88. Notice that the set of choices for d1d_1 and d2d_2 are disjoint.
3
Determine the number of valid choices for the third digit (d3d_3) and fourth digit (d4d_4)
6 choices for d3d_3 and 5 choices for d4d_4
Since all 4 digits must be distinct, 2 distinct digits have already been used for d1d_1 and d2d_2. From the original set of 8 digits, 8−2=68 - 2 = 6 digits remain for d3d_3, and then 55 digits remain for d4d_4.
4
Apply the Fundamental Counting Principle to calculate the total number of codes
3×4×6×5=3603 \times 4 \times 6 \times 5 = 360
The overall number of distinct multi-stage decisions is found by multiplying the number of choices available at each stage.

Key Concept

Fundamental Counting Principle with Restricted and Non-Repeating Selections
Estimated Time:2m 0s
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Topics

Fundamental Counting Principle

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