The number of five - digit telephone numbers having atleast one of their digits repeated is (00000 is also a telephone number)

Options

A. 90000
B. 10000
C. 30240
D. 69760

Answer

To solve the problem of finding the number of five-digit telephone numbers that have at least one digit repeated, we can follow these steps:

### Step 1: Calculate the total number of five-digit telephone numbers
Since a five-digit telephone number can include the digit 0 and can have any digit from 0 to 9, we have 10 choices for each of the 5 digits.

\[ \text{Total numbers} = 10^5 = 100000 \]

### Step 2: Calculate the number of five-digit telephone numbers with no repeated digits
To find the number of five-digit telephone numbers where no digits are repeated, we can follow this approach:

1. For the first digit, we can choose any of the 10 digits (0-9).
2. For the second digit, we can choose from the remaining 9 digits (since one digit has already been used).
3. For the third digit, we can choose from the remaining 8 digits.
4. For the fourth digit, we can choose from the remaining 7 digits.
5. For the fifth digit, we can choose from the remaining 6 digits.

Thus, the calculation will be:

\[ \text{Total numbers with no repeated digits} = 10 \times 9 \times 8 \times 7 \times 6 \]

Calculating this gives:

\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]
\[ 720 \times 7 = 5040 \]
\[ 5040 \times 6 = 30240 \]

So, the total number of five-digit telephone numbers with no repeated digits is 30240.

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

\[ \text{Numbers with at least one repeated digit} = \text{Total numbers} - \text{Numbers with no repeated digits} \]
\[ = 100000 - 30240 = 69760 \]

### Final Answer
The number of five-digit telephone numbers having at least one of their digits repeated is **69,760**.

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