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Difficulty: EasyPermutations, Combinations, and Fundamental Counting Principle

A security code consists of three distinct digits chosen from the non-zero digits 11 through 99. If the first digit must be odd and the third digit must be even, how many such three-digit security codes can be formed?

  1. A
    16
  2. B
    70
  3. 140Answer
  4. D
    160
  5. E
    180

Answer

140
To find the number of three-digit codes with distinct digits from 11 through 99 satisfying the constraints, count the options for each slot: the first position has 55 odd options (1,3,5,7,91, 3, 5, 7, 9), the third position has 44 even options (2,4,6,82, 4, 6, 8), and the middle position has 9−2=79 - 2 = 7 remaining options. By the Fundamental Counting Principle, multiplying these options gives 5×7×4=1405 \times 7 \times 4 = 140.

Step-by-Step Solution

1
Determine the number of possibilities for the first digit.
5 choices (the odd digits: 1,3,5,7,91, 3, 5, 7, 9).
The question specifies that the first digit must be odd.
2
Determine the number of possibilities for the third digit.
4 choices (the even digits: 2,4,6,82, 4, 6, 8).
The question specifies that the third digit must be even.
3
Determine the number of possibilities for the middle (second) digit.
7 choices.
There are 99 total non-zero digits (11 through 99). Since 22 distinct digits have already been used for the first and third positions, 9−2=79 - 2 = 7 digits remain available for the middle position.
4
Apply the Fundamental Counting Principle to find the total number of codes.
5×7×4=1405 \times 7 \times 4 = 140.
The total number of sequential independent choices is found by multiplying the number of options for each position.

Key Concept

Fundamental Counting Principle with Restricted Positions and Distinct Elements
Estimated Time:45s
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Topics

Permutations, Combinations, and Fundamental Counting Principle

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