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Difficulty: HardCounting Principles

A 5-character security passcode must be created using the digits 0 through 9. The passcode must begin with an odd digit, end with an even digit, and no digit may be repeated anywhere in the passcode. How many different passcodes can be created?

  1. A
    1,400
  2. 8,400Answer
  3. C
    18,000
  4. D
    25,000

Answer

8,400
To find the total number of possible passcodes, we evaluate the restricted positions first. The first digit must be odd (5 options: 1, 3, 5, 7, 9). The last digit must be even (5 options: 0, 2, 4, 6, 8). Because odd and even sets do not overlap, exactly one odd and one even digit are removed from the total pool of 10 digits. Because repetition is not allowed, 10−2=810 - 2 = 8 digits remain available for the middle three positions. The number of ways to fill the middle three spots is a permutation without repetition: 8×7×6=3368 \times 7 \times 6 = 336. Multiplying the independent choices for all positions gives 5×336×5=8,4005 \times 336 \times 5 = 8,400.

Step-by-Step Solution

1
Determine the number of options for the first character.
5 options
The first digit must be odd, and there are 5 odd digits (1, 3, 5, 7, 9) in the standard set of digits from 0-9.
2
Determine the number of options for the last character.
5 options
The last digit must be even, and there are 5 even digits (0, 2, 4, 6, 8) in the standard set.
3
Determine the number of available digits for the remaining three middle positions.
8 digits
There are 10 digits in total. Since no digit may be repeated and 2 distinct digits have already been used for the first and last positions, exactly 8 digits remain available.
4
Calculate the number of ways to arrange the middle three characters.
336 ways
Using permutations for the 3 middle spots from the 8 remaining digits: 8 × 7 × 6 = 336.
5
Calculate the total number of possible passcodes.
8,400 passcodes
Using the Fundamental Counting Principle, multiply the number of options for each stage: 5 × 336 × 5 = 8,400.

Key Concept

Fundamental Counting Principle with Restrictions
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Topics

Counting Principles

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