Question

The number of four-digit numbers that can be formed from the digits 0, 1, 2, 3, 4, 5 with at least one digit repeated is:

• A. 420
• B. 560
• C. 780
• D. none of the above

Hint:

To find the number of four-digit numbers with at least one repeated digit, first calculate the total number of possible numbers, then subtract the number of numbers with no repetition.

Answer 1

The Correct Option is B

We are given the digits \( \{0, 1, 2, 3, 4, 5\} \) and need to find the number of four-digit numbers that can be formed with at least one digit repeated. 1. Total number of four-digit numbers: A four-digit number cannot start with 0, so we have 5 choices for the first digit (1, 2, 3, 4, or 5), and 6 choices for each of the other three digits (0, 1, 2, 3, 4, or 5): \[ \text{Total numbers} = 5 \times 6 \times 6 \times 6 = 1080 \] 2. Number of four-digit numbers with no digits repeated: For a four-digit number with no digits repeated, we have 5 choices for the first digit (1, 2, 3, 4, or 5), 5 choices for the second digit (remaining 5 digits), 4 choices for the third digit, and 3 choices for the fourth digit: \[ \text{Numbers with no repetition} = 5 \times 5 \times 4 \times 3 = 300 \] 3. Number of four-digit numbers with at least one digit repeated: To find this, we subtract the number of numbers with no repetition from the total number of numbers: \[ \text{Numbers with repetition} = 1080 - 300 = 780 \] Thus, the correct answer is \( \boxed{780} \).