Five digit numbers are formed with 0 1 2 3 4 The number of numbers in which at least one digit is repeated is

Options

A. 96
B. 120
C. 2500
D. 2404

Answer

To find the number of five-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4, where at least one digit is repeated, we can follow these steps:

### Step 1: Calculate the Total Number of Five-Digit Numbers
To form a five-digit number, the first digit cannot be 0 (to ensure it's a five-digit number). Therefore, we have the following choices:

- **First digit**: 4 options (1, 2, 3, 4)
- **Second digit**: 5 options (0, 1, 2, 3, 4)
- **Third digit**: 5 options (0, 1, 2, 3, 4)
- **Fourth digit**: 5 options (0, 1, 2, 3, 4)
- **Fifth digit**: 5 options (0, 1, 2, 3, 4)

Thus, the total number of five-digit numbers is calculated as follows:

\[ \text{Total five-digit numbers} = 4 \times 5 \times 5 \times 5 \times 5 = 4 \times 625 = 2500 \]

### Step 2: Calculate the Number of Five-Digit Numbers with All Unique Digits
Next, we calculate the number of five-digit numbers where all digits are unique. The first digit again cannot be 0, so we have:

- **First digit**: 4 options (1, 2, 3, 4)
- **Second digit**: 4 options (remaining digits including 0)
- **Third digit**: 3 options (remaining digits)
- **Fourth digit**: 2 options (remaining digits)
- **Fifth digit**: 1 option (remaining digit)

Thus, the total number of five-digit numbers with all unique digits is:

\[ \text{Unique five-digit numbers} = 4 \times 4 \times 3 \times 2 \times 1 = 96 \]

### Step 3: Calculate the Number of Five-Digit Numbers with At Least One Repeated Digit
To find the number of five-digit numbers with at least one digit repeated, we subtract the number of unique five-digit numbers from the total five-digit numbers:

\[ \text{At least one repeated digit} = \text{Total five-digit numbers} - \text{Unique five-digit numbers} \]

Substituting the values we calculated:

\[ \text{At least one repeated digit} = 2500 - 96 = 2404 \]

### Final Answer
The number of five-digit numbers formed with the digits 0, 1, 2, 3, and 4, in which at least one digit is repeated, is **2404**.

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