Examkin

Question

Difficulty: HardFundamental Counting Principle

A security analyst is designing a 4-character access code consisting of a digit, followed by two letters, followed by another digit. The code must be constructed according to the following rules:

- The first character must be a prime digit chosen from the set of single-digit integers {0,1,2,3,4,5,6,7,8,9}\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}.
- The second and third characters must be distinct uppercase letters selected from the set {A,B,C,D,E}\{A, B, C, D, E\}.
- The fourth character must be an odd digit chosen from the set of single-digit integers {0,1,2,3,4,5,6,7,8,9}\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}, such that it is not equal to the first digit.

How many different 4-character access codes can be created following these rules?

  1. A
    300
  2. 340Answer
  3. C
    400
  4. D
    420
  5. E
    500

Answer

340 access codes
The correct count of 340 is obtained by separating the problem into two distinct cases based on whether the prime first digit is even or odd. For the even prime digit (2), there are 5 possible odd digits for the fourth slot, yielding 100 codes. For the three odd prime digits (3, 5, 7), the fourth slot has 4 available choices to avoid duplicating the first digit, yielding 240 codes. Summing both cases gives 340.

Step-by-Step Solution

1
Identify the set of prime single-digit integers.
The prime digits among 00 through 99 are 2,3,5,2, 3, 5, and 77 (a total of 44 options). Note that 22 is even, while 3,5,3, 5, and 77 are odd.
11 is not prime by definition, and 0,4,6,8,90, 4, 6, 8, 9 are composite or neither.
2
Calculate the number of ways to choose the two distinct letter characters.
5×4=205 \times 4 = 20 possible two-letter arrangements.
There are 55 choices for the second character and 44 remaining choices for the third character since they must be distinct.
3
Evaluate Case 1: The first digit is the even prime digit (22).
1×20×5=1001 \times 20 \times 5 = 100 codes.
There is 11 choice for the first digit (22). The odd digits available for the fourth character are 1,3,5,7,91, 3, 5, 7, 9 (55 choices). Since 22 is even, it never matches any odd digit.
4
Evaluate Case 2: The first digit is an odd prime digit (3,5,3, 5, or 77).
3×20×4=2403 \times 20 \times 4 = 240 codes.
There are 33 choices for the first digit. The odd digits available for the fourth character are 1,3,5,7,91, 3, 5, 7, 9 (55 total), but the fourth digit cannot equal the first digit, leaving 5−1=45 - 1 = 4 choices.
5
Sum the total codes across both mutually exclusive cases.
100+240=340100 + 240 = 340 total codes.
By the Fundamental Counting Principle and Addition Principle for disjoint sets.

Key Concept

Fundamental Counting Principle with Conditional Restrictions and Case Analysis
Estimated Time:2m 0s
Rate this question

Topics

Fundamental Counting Principle

More questions from this topic