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

Options

A. 90000
B. 100000
C. 27216
D. 62784

Answer

To find the number of 5-digit numbers that have at least one digit repeated, we can use the principle of complementary counting. Here's a step-by-step solution:

### Step 1: Calculate the total number of 5-digit numbers
A 5-digit number cannot start with 0. Therefore, the first digit can be any digit from 1 to 9 (9 options). The remaining four digits can be any digit from 0 to 9 (10 options each).

Total number of 5-digit numbers =
\[ 9 \times 10 \times 10 \times 10 \times 10 = 9 \times 10^4 = 90,000 \]

### Step 2: Calculate the number of 5-digit numbers with no repeated digits
For a 5-digit number with no repeated digits:
- The first digit can be any digit from 1 to 9 (9 options).
- The second digit can be any digit from 0 to 9 except the first digit (9 options).
- The third digit can be any digit from 0 to 9 except the first two digits (8 options).
- The fourth digit can be any digit from 0 to 9 except the first three digits (7 options).
- The fifth digit can be any digit from 0 to 9 except the first four digits (6 options).

Total number of 5-digit numbers with no repeated digits =
\[ 9 \times 9 \times 8 \times 7 \times 6 \]

Calculating this:
\[ 9 \times 9 = 81 \]
\[ 81 \times 8 = 648 \]
\[ 648 \times 7 = 4536 \]
\[ 4536 \times 6 = 27216 \]

So, the total number of 5-digit numbers with no repeated digits is 27,216.

### Step 3: Calculate the number of 5-digit numbers with at least one digit repeated
To find the number of 5-digit numbers that have at least one digit repeated, we subtract the number of 5-digit numbers with no repeated digits from the total number of 5-digit numbers.

Number of 5-digit numbers with at least one digit repeated =
\[ 90,000 - 27,216 = 62,784 \]

### Final Answer:
The number of 5-digit numbers having at least one digit repeated is **62,784**.

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