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Difficulty: MediumPermutations, Combinations, and Fundamental Counting Principle

A security system requires a 4-digit pass code formed using the digits 1,2,3,4,5,6,1, 2, 3, 4, 5, 6, and 77, with no digit repeated within a code. If the first digit of the pass code must be an even number and the last digit must be an odd number, how many such distinct pass codes can be created?

  1. A
    120
  2. 240Answer
  3. C
    300
  4. D
    504
  5. E
    840

Answer

240 pass codes
To form a 4-digit code with distinct digits from the set {1, 2, 3, 4, 5, 6, 7}: there are 3 options for the first digit (even: 2, 4, 6) and 4 options for the fourth digit (odd: 1, 3, 5, 7). Because the sets of even and odd numbers are disjoint, choosing the first digit does not affect the number of odd choices available for the fourth position. After placing these 2 digits, 5 digits remain from the original set of 7. The second position can be filled in 5 ways, and the third position in 4 ways. By the Fundamental Counting Principle, the total number of codes is 3 × 4 × 5 × 4 = 240.

Step-by-Step Solution

1
Identify the choices for the first digit (even restriction).
3 possible choices (2, 4, or 6).
The first digit must be even, and the available set contains three even digits: {2, 4, 6}.
2
Identify the choices for the last (fourth) digit (odd restriction).
4 possible choices (1, 3, 5, or 7).
The fourth digit must be odd, and the available set contains four odd digits: {1, 3, 5, 7}.
3
Determine the available choices for the remaining middle digits.
5 choices for the second digit and 4 choices for the third digit.
Two distinct digits have been selected for the first and fourth positions out of the 7 available digits, leaving 5 remaining digits. Since no digits may repeat, the second position has 5 choices and the third position has 4 choices.
4
Apply the Fundamental Counting Principle to compute the total number of pass codes.
3 × 4 × 5 × 4 = 240 distinct pass codes.
Multiplying the independent choices for each position gives the total valid arrangements.

Key Concept

Fundamental Counting Principle and Permutations with Positional Restrictions
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Topics

Permutations, Combinations, and Fundamental Counting Principle

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