Consider all `10` digit numbers formed by using all the digits 0, 1, 2, 3,…, 9 without repetition such that they are divisible by 11111, then

Options

A. the digit in tens place for smallest number is 6
B. the digit in tens place for largest number is 3
C. total numbers of such numbers is 3456
D. total numbers of such numbers is 365

Explanation

To solve the problem of finding all 10-digit numbers formed by using the digits 0 to 9 without repetition that are divisible by 11111, we can break the solution down into several steps.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find 10-digit numbers formed by the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 without repetition, such that the number is divisible by 11111.

2. **Divisibility Condition**:
A number is divisible by 11111 if it is also divisible by 9 (since 11111 = 9 * 1234 + 5). Therefore, we need to check if our 10-digit number is divisible by 9.

3. **Sum of Digits**:
The sum of all digits from 0 to 9 is:
\[ 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \]
Since 45 is divisible by 9, any arrangement of these digits will also be divisible by 9.

4. **Forming the Number**:
We can represent our 10-digit number as \( N = x_1 x_2 x_3 x_4 x_5 x_6 x_7 x_8 x_9 x_{10} \), where each \( x_i \) is a digit from 0 to 9.

5. **Finding the Smallest and Largest Numbers**:
- **Smallest Number**: To find the smallest number, we arrange the digits in ascending order, ensuring that the first digit is not 0. The smallest valid arrangement is 1023456789.
- **Largest Number**: To find the largest number, we arrange the digits in descending order, which gives us 9876543210.

6. **Checking the 10th Place Digit**:
- In the smallest number (1023456789), the digit in the 10th place is 9.
- In the largest number (9876543210), the digit in the 10th place is 0.

7. **Counting the Total Numbers**:
The total number of 10-digit numbers that can be formed using all digits from 0 to 9 without repetition is given by the factorial of the number of digits:
\[ 10! = 3628800 \]

### Conclusion:
The smallest number is 1023456789, the largest number is 9876543210, and the total number of valid 10-digit numbers is 3628800.