4-Digit Even Numbers Problem
Problem
Find the number of 4-digit even numbers that can be formed using the digits {0, 1, 2, 3, 4, 5}, where each digit is used at most once in each number.
Solution
To form a 4-digit even number with distinct digits from {0, 1, 2, 3, 4, 5}, the last digit must be even (0, 2, or 4), the first digit cannot be 0, and all digits must be unique. We calculate using a case-by-case approach based on the last digit:
- Case 1: Last digit is 0
- Units place: 0 (1 choice)
- Thousands place: {1, 2, 3, 4, 5} (5 choices)
- Hundreds place: 4 remaining digits (4 choices)
- Tens place: 3 remaining digits (3 choices)
- Total: 1 × 5 × 4 × 3 = 60 numbers
- Case 2: Last digit is 2
- Units place: 2 (1 choice)
- Thousands place: {1, 3, 4, 5} (4 choices, excluding 0)
- Hundreds place: 4 remaining digits (4 choices)
- Tens place: 3 remaining digits (3 choices)
- Total: 1 × 4 × 4 × 3 = 48 numbers
- Case 3: Last digit is 4
- Units place: 4 (1 choice)
- Thousands place: {1, 2, 3, 5} (4 choices, excluding 0)
- Hundreds place: 4 remaining digits (4 choices)
- Tens place: 3 remaining digits (3 choices)
- Total: 1 × 4 × 4 × 3 = 48 numbers
Summing the cases: 60 + 48 + 48 = 156.
Therefore, the number of 4-digit even numbers is 156.