The number of 5 digit telephone numbers having atleast one of their digits repeated is ?

To determine the number of 5-digit telephone numbers that have at least one digit repeated, it’s often easier to first calculate the total number of 5-digit combinations and then subtract the number of combinations where all digits are unique. Let’s break this down step by step.

Understanding the Total Combinations of 5-Digit Numbers

A 5-digit telephone number can range from 00000 to 99999. This means that each digit can be any number from 0 to 9. Therefore, the total number of 5-digit combinations is:

• 10 options (0-9) for the first digit
• 10 options for the second digit
• 10 options for the third digit
• 10 options for the fourth digit
• 10 options for the fifth digit

So, the total number of 5-digit telephone numbers is:

Total = 10 × 10 × 10 × 10 × 10 = 100,000

Calculating Unique Digit Combinations

Next, let’s find the number of 5-digit combinations where all digits are unique. In this case, we consider the following:

• For the first digit, there are 10 choices (0-9).
• For the second digit, we can only choose from the remaining 9 digits.
• For the third digit, we can choose from the remaining 8 digits.
• For the fourth digit, we have 7 choices left.
• For the fifth digit, we can choose from the remaining 6 digits.

The calculation would look like this:

Unique combinations = 10 × 9 × 8 × 7 × 6

Now, let’s perform that calculation:

Unique combinations = 10 × 9 × 8 × 7 × 6 = 30,240

Finding the Numbers with Repeated Digits

To find the number of 5-digit telephone numbers that have at least one digit repeated, we subtract the number of unique combinations from the total combinations:

Numbers with at least one repeated digit = Total combinations - Unique combinations

Numbers with at least one repeated digit = 100,000 - 30,240 = 69,760

Final Result

Thus, the total number of 5-digit telephone numbers that have at least one digit repeated is 69,760.

This method highlights how breaking down problems into manageable parts can clarify your understanding of combinatorial mathematics.