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Difficulty: HardFundamental Counting Principle

A tech company generates a 5-digit security badge number, represented as d1d2d3d4d5d_1 d_2 d_3 d_4 d_5, using digits from 00 to 99, inclusive. The badge numbers must satisfy the following constraints:
- The first digit, d1d_1, cannot be 00 or 11.
- The second and third digits, d2d_2 and d3d_3, must be distinct even digits.
- The fourth digit, d4d_4, must be an odd digit strictly greater than 33.
- The fifth digit, d5d_5, can be any digit except that it cannot be equal to d1d_1.

How many different 5-digit badge numbers can be created under these rules?

  1. 4,320Answer
  2. B
    4,860
  3. C
    5,400
  4. D
    5,760
  5. E
    6,000

Answer

4,320
According to the Fundamental Counting Principle, the total number of configurations is the product of the number of options available at each stage. For the first position, excluding 0 and 1 leaves 8 possible digits. For the second position, any of the 5 even digits can be chosen. For the third position, one of the remaining 4 even digits must be selected to preserve distinctness. For the fourth position, the odd digits strictly greater than 3 are 5, 7, and 9, providing 3 options. Finally, for the fifth position, 9 digits remain available after excluding the specific digit chosen for the first position. Multiplying these independent counts gives 8 × 5 × 4 × 3 × 9 = 4,320.

Step-by-Step Solution

1
Determine the number of available choices for the first digit, d1d_1.
8 choices
The available digits are 00 through 99 (10 total). Excluding 00 and 11 leaves {2,3,4,5,6,7,8,9}\{2, 3, 4, 5, 6, 7, 8, 9\}, which gives 10−2=810 - 2 = 8 choices.
2
Determine the number of available choices for the second and third digits, d2d_2 and d3d_3.
20 choices total for (d2,d3)(d_2, d_3)
The set of even digits is {0,2,4,6,8}\{0, 2, 4, 6, 8\}, containing 5 digits. There are 5 choices for d2d_2. Since d3d_3 must be distinct from d2d_2, there are 5−1=45 - 1 = 4 choices for d3d_3. Thus, there are 5×4=205 \times 4 = 20 choices.
3
Determine the number of available choices for the fourth digit, d4d_4.
3 choices
The set of odd digits is {1,3,5,7,9}\{1, 3, 5, 7, 9\}. Odd digits strictly greater than 33 are {5,7,9}\{5, 7, 9\}, which gives 3 choices.
4
Determine the number of available choices for the fifth digit, d5d_5.
9 choices
d5d_5 can be any digit from 00 through 99 except d1d_1. Out of 10 total digits, excluding 1 digit leaves 10−1=910 - 1 = 9 choices.
5
Apply the Fundamental Counting Principle to compute total combinations.
4,320
Multiply the number of independent choices across all 5 stages: 8×5×4×3×9=4,3208 \times 5 \times 4 \times 3 \times 9 = 4,320.

Key Concept

Fundamental Counting Principle with Multi-Stage Positional Restrictions
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Topics

Fundamental Counting Principle

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