The number of 6 digits numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is

Options

A. 60
B. 72
C. 48
D. 36

Answer

To solve the problem of finding the number of 6-digit numbers that can be formed using the digits 0, 1, 2, 5, 7, and 9, which are divisible by 11 and have no repeated digits, we can follow these steps:

### Step 1: Understand the divisibility rule for 11
A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11.

### Step 2: Identify the digits and their sum
The digits available are 0, 1, 2, 5, 7, and 9. The total sum of these digits is:
\[ 0 + 1 + 2 + 5 + 7 + 9 = 24 \]

### Step 3: Set up the equation for divisibility by 11
Let’s denote the digits in the 6-digit number as \( a, b, c, d, e, f \). According to the rule for divisibility by 11:
\[ (a + c + e) - (b + d + f) \equiv 0 \ (\text{mod } 11) \]
This can also be expressed as:
\[ a + c + e = b + d + f \]
Since the total sum of the digits is 24, we can set:
\[ a + c + e = b + d + f = 12 \]

### Step 4: Consider possible combinations for \( a, c, e \) and \( b, d, f \)
We need to find combinations of three digits that sum to 12.

#### Case 1: \( a, c, e = 9, 2, 1 \)
- The remaining digits \( b, d, f \) will be \( 0, 5, 7 \).
- The number of arrangements for \( a, c, e \) is \( 3! = 6 \).
- The number of arrangements for \( b, d, f \) is also \( 3! = 6 \).
- Total arrangements for Case 1:
\[ 6 \times 6 = 36 \]

#### Case 2: \( a, c, e = 7, 5, 0 \)
- The remaining digits \( b, d, f \) will be \( 9, 2, 1 \).
- Since 0 cannot be the leading digit, we have 2 choices for the first digit (7 or 5), and the arrangements will be \( 2 \times 2! = 4 \).
- The arrangements for \( b, d, f \) is \( 3! = 6 \).
- Total arrangements for Case 2:
\[ 4 \times 6 = 24 \]

### Step 5: Calculate the total number of valid 6-digit numbers
Now, we sum the valid arrangements from both cases:
\[ 36 + 24 = 60 \]

### Final Answer
The total number of 6-digit numbers that can be formed is **60**.

---