The number of five digit numbers having atleast one of their digits repeated is :

Options

A. 900
B. 1000
C. 62784
D. none of these

Answer

To find the number of five-digit numbers having at least one digit repeated, we can follow these steps:

### Step 1: Calculate the total number of five-digit numbers
A five-digit number cannot start with 0. Therefore, the first digit can be any of the digits from 1 to 9 (9 options). The remaining four digits can be any digit from 0 to 9 (10 options each).

- **First digit:** 9 options (1-9)
- **Second digit:** 10 options (0-9)
- **Third digit:** 10 options (0-9)
- **Fourth digit:** 10 options (0-9)
- **Fifth digit:** 10 options (0-9)

Thus, the total number of five-digit numbers is calculated as:
\[ \text{Total five-digit numbers} = 9 \times 10^4 = 90,000 \]

### Step 2: Calculate the total number of five-digit numbers with no repeated digits
To find the number of five-digit numbers with no repeated digits, we consider the following:

- **First digit:** 9 options (1-9)
- **Second digit:** 9 options (0-9 excluding the first digit)
- **Third digit:** 8 options (0-9 excluding the first two digits)
- **Fourth digit:** 7 options (0-9 excluding the first three digits)
- **Fifth digit:** 6 options (0-9 excluding the first four digits)

Thus, the total number of five-digit numbers with no repeated digits is calculated as:
\[ \text{Total non-repeating five-digit numbers} = 9 \times 9 \times 8 \times 7 \times 6 \]

Calculating this step-by-step:
1. \( 9 \times 9 = 81 \)
2. \( 81 \times 8 = 648 \)
3. \( 648 \times 7 = 4536 \)
4. \( 4536 \times 6 = 27216 \)

So, the total number of five-digit numbers with no repeated digits is \( 27,216 \).

### Step 3: Calculate the number of five-digit numbers with at least one digit repeated
To find the number of five-digit numbers that have at least one digit repeated, we subtract the total number of five-digit numbers with no repeated digits from the total number of five-digit numbers.

\[ \text{Five-digit numbers with at least one digit repeated} = \text{Total five-digit numbers} - \text{Total non-repeating five-digit numbers} \]
\[ = 90,000 - 27,216 = 62,784 \]

### Final Answer
The number of five-digit numbers having at least one of their digits repeated is **62,784**.

---