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Difficulty: Very hardBasic Probability and Counting Methods

A security passcode consists of a sequence of 4 digits chosen from the digits 0 through 9. To be valid, a passcode must contain at least one repeated digit, but no digit can appear more than twice. Additionally, the passcode cannot end with an odd digit. How many different valid security passcodes can be created?

Answer: 2295

Answer

There are 2,295 different valid security passcodes.
The correct answer of 2,295 is found by subtracting all invalid passcodes from the total possible passcodes. The total number of 4-digit passcodes ending in an even digit is 5,000. The invalid passcodes are those with all distinct digits (2,520), those with a digit repeated three times (180), and those with all four digits identical (5). Subtracting these gives 5,000 - 2,520 - 180 - 5 = 2,295.

Step-by-Step Solution

1
Calculate the total number of 4-digit passcodes ending in an even digit.
5,000
The last digit must be even (0, 2, 4, 6, or 8) to not be odd, giving 5 choices. The first three digits can be any of the 10 digits from 0 through 9. By the Fundamental Counting Principle, the total number of passcodes is 10 * 10 * 10 * 5 = 5,000.
2
Calculate the number of invalid passcodes where all 4 digits are distinct.
2,520
For all 4 digits to be distinct, the last digit must be chosen from the 5 even digits. The remaining 3 positions must be filled with 3 distinct digits chosen from the remaining 9 digits. There are 9 * 8 * 7 = 504 ways to choose these. This gives 504 * 5 = 2,520 passcodes.
3
Calculate the number of invalid passcodes where a single digit is repeated three times.
180
If the repeated digit is the last digit, there are 5 choices for that digit, and the single distinct digit (9 choices) can be placed in any of the 3 remaining positions, giving 5 * 9 * 3 = 135 passcodes. If the repeated digit is not the last digit, the three identical digits occupy the first three positions, and the last digit (5 choices) is distinct from them (9 choices), giving 5 * 9 = 45 passcodes. In total, 135 + 45 = 180 passcodes.
4
Calculate the number of invalid passcodes where all four digits are identical.
5
Since the last digit must be even, all four digits must be the same even digit (0000, 2222, 4444, 6666, or 8888), which gives 5 passcodes.
5
Subtract the invalid passcodes from the total number of passcodes.
2,295
Subtracting the passcodes with all distinct digits (2,520) and those with a digit repeated three or four times (180 + 5 = 185) from the total of 5,000 gives 5,000 - 2,520 - 185 = 2,295.

Key Concept

Complementary counting using permutations and partition analysis

Alternative Method

Instead of complementary counting, count the valid cases directly: Case A where the digit frequencies are [2, 1, 1] (which yields 2,160 codes) and Case B where the digit frequencies are [2, 2] (which yields 135 codes). Adding these yields 2,160 + 135 = 2,295 codes.
Estimated Time:3m 0s
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Topics

Basic Probability and Counting Methods

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