The number of three digits numbers formed by the number 2, 3, 4, 5, 7 without repetition which are not divisible by 3 is

Options

A. 24
B. 36
C. 12
D. 48

Answer

To solve the problem of finding the number of three-digit numbers formed by the digits 2, 3, 4, 5, and 7 without repetition that are not divisible by 3, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Digits**:
We have the digits 2, 3, 4, 5, and 7. This gives us a total of 5 digits.

2. **Calculate Total Three-Digit Numbers**:
To form a three-digit number without repetition:
- We can choose any of the 5 digits for the hundreds place.
- For the tens place, we can choose from the remaining 4 digits.
- For the units place, we can choose from the remaining 3 digits.

Therefore, the total number of three-digit numbers is:
\[ 5 \times 4 \times 3 = 60 \]

3. **Identify Numbers Divisible by 3**:
A number is divisible by 3 if the sum of its digits is divisible by 3. We need to find combinations of three digits from our set that sum up to a number divisible by 3.

- The digits are: 2, 3, 4, 5, 7.
- Their remainders when divided by 3 are:
- 2 % 3 = 2
- 3 % 3 = 0
- 4 % 3 = 1
- 5 % 3 = 2
- 7 % 3 = 1
- The remainders are: 2, 0, 1, 2, 1.

We can find combinations of three digits whose sum is divisible by 3. The combinations are:
- (2, 3, 4) → 2 + 3 + 4 = 9 (divisible by 3)
- (2, 3, 5) → 2 + 3 + 5 = 10 (not divisible by 3)
- (2, 3, 7) → 2 + 3 + 7 = 12 (divisible by 3)
- (2, 4, 5) → 2 + 4 + 5 = 11 (not divisible by 3)
- (2, 4, 7) → 2 + 4 + 7 = 13 (not divisible by 3)
- (2, 5, 7) → 2 + 5 + 7 = 14 (not divisible by 3)
- (3, 4, 5) → 3 + 4 + 5 = 12 (divisible by 3)
- (3, 4, 7) → 3 + 4 + 7 = 14 (not divisible by 3)
- (3, 5, 7) → 3 + 5 + 7 = 15 (divisible by 3)
- (4, 5, 7) → 4 + 5 + 7 = 16 (not divisible by 3)

The valid combinations that are divisible by 3 are:
- (2, 3, 4)
- (2, 3, 7)
- (3, 4, 5)
- (3, 5, 7)

This gives us a total of 4 combinations.

4. **Calculate Numbers Divisible by 3**:
Each combination can form \(3!\) (which is 6) different three-digit numbers since the digits can be arranged in any order. Therefore, the total number of three-digit numbers that are divisible by 3 is:
\[ 4 \times 6 = 24 \]

5. **Calculate Numbers Not Divisible by 3**:
To find the numbers that are not divisible by 3, we subtract the numbers that are divisible by 3 from the total three-digit numbers:
\[ 60 - 24 = 36 \]

### Final Answer:
The number of three-digit numbers formed by the digits 2, 3, 4, 5, and 7 without repetition that are not divisible by 3 is **36**.