The total number of numbers greater than 100 and divisible by 5, that can be formed from the digits 3, 4, 5, 6 if no digit is repeated, is

Options

A. 24
B. 48
C. 30
D. 12

Answer

To solve the problem of finding the total number of numbers greater than 100 and divisible by 5 that can be formed from the digits 3, 4, 5, and 6 (with no digit repeated), we can break it down into steps.

### Step-by-Step Solution:

**Step 1: Understand the conditions**
- The numbers must be greater than 100.
- The numbers must be divisible by 5.
- The digits available are 3, 4, 5, and 6.
- No digit can be repeated.

**Step 2: Determine the last digit for divisibility by 5**
- A number is divisible by 5 if its last digit is either 0 or 5. Since we do not have 0 in our digits, the last digit must be 5.

**Step 3: Count three-digit numbers**
- For a three-digit number, the last digit is fixed as 5.
- The first digit can be either 3, 4, or 6 (it must be greater than 0).
- The second digit can be any of the remaining digits after choosing the first digit.

Calculating the number of three-digit numbers:
- **Choices for the first digit**: 3 options (3, 4, or 6)
- **Choices for the second digit**: 2 options (remaining digits after choosing the first)

Thus, the number of three-digit numbers is:
\[ 3 \text{ (choices for first digit)} \times 2 \text{ (choices for second digit)} = 6 \]

**Step 4: Count four-digit numbers**
- For a four-digit number, the last digit is again fixed as 5.
- The first digit can be either 3, 4, or 6.
- The second and third digits can be chosen from the remaining two digits.

Calculating the number of four-digit numbers:
- **Choices for the first digit**: 3 options (3, 4, or 6)
- **Choices for the second digit**: 2 options (remaining digits after choosing the first)
- **Choices for the third digit**: 1 option (the last remaining digit)

Thus, the number of four-digit numbers is:
\[ 3 \text{ (choices for first digit)} \times 2 \text{ (choices for second digit)} \times 1 \text{ (choice for third digit)} = 6 \]

**Step 5: Calculate the total**
- Total numbers greater than 100 and divisible by 5:
\[ 6 \text{ (three-digit numbers)} + 6 \text{ (four-digit numbers)} = 12 \]

### Final Answer:
The total number of numbers greater than 100 and divisible by 5 that can be formed from the digits 3, 4, 5, and 6 is **12**.