The number of all five digit numbers which are divisible by 4 that can be formed from the digits 0,1,2,3,4 (without repetition) is a. 36 b. 30 c. 34 d. None of these

Options

A. 36
B. 30
C. 34
D. None of these

Answer

To solve the problem of finding the number of all five-digit numbers that can be formed from the digits 0, 1, 2, 3, and 4 (without repetition) and are divisible by 4, we need to follow these steps:

### Step 1: Understand the divisibility rule for 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Therefore, we need to consider all possible pairs of the last two digits that can be formed from the given digits (0, 1, 2, 3, 4) and check which of these pairs are divisible by 4.

### Step 2: Identify valid pairs of last two digits
The possible pairs of last two digits that can be formed from the digits 0, 1, 2, 3, and 4 are:
- 04
- 12
- 20
- 24
- 32
- 40

Now, we will check which of these pairs are divisible by 4:
- 04 is divisible by 4
- 12 is divisible by 4
- 20 is divisible by 4
- 24 is divisible by 4
- 32 is divisible by 4
- 40 is divisible by 4

All pairs listed above are valid.

### Step 3: Count the valid five-digit numbers for each pair
We will analyze each pair of last two digits and count how many valid five-digit numbers can be formed:

1. **Last two digits = 04**
- Remaining digits: 1, 2, 3
- The first digit cannot be 0 (it must be a five-digit number).
- Choices for the first digit: 1, 2, or 3 (3 options).
- The remaining two positions can be filled with the remaining two digits (2 options).
- Total combinations = 3 (first digit) × 2! (arrangements of the remaining) = 3 × 2 = 6.

2. **Last two digits = 12**
- Remaining digits: 0, 3, 4
- Choices for the first digit: 3 or 4 (2 options).
- Total combinations = 2 (first digit) × 2! = 2 × 2 = 4.

3. **Last two digits = 20**
- Remaining digits: 1, 3, 4
- Choices for the first digit: 1, 3, or 4 (3 options).
- Total combinations = 3 × 2! = 3 × 2 = 6.

4. **Last two digits = 24**
- Remaining digits: 0, 1, 3
- Choices for the first digit: 1 or 3 (2 options).
- Total combinations = 2 × 2! = 2 × 2 = 4.

5. **Last two digits = 32**
- Remaining digits: 0, 1, 4
- Choices for the first digit: 1 or 4 (2 options).
- Total combinations = 2 × 2! = 2 × 2 = 4.

6. **Last two digits = 40**
- Remaining digits: 1, 2, 3
- Choices for the first digit: 1, 2, or 3 (3 options).
- Total combinations = 3 × 2! = 3 × 2 = 6.

### Step 4: Add all the combinations
Now, we add the total combinations from all cases:
- From 04: 6
- From 12: 4
- From 20: 6
- From 24: 4
- From 32: 4
- From 40: 6

Total = 6 + 4 + 6 + 4 + 4 + 6 = 30

### Final Answer
Thus, the total number of five-digit numbers that can be formed from the digits 0, 1, 2, 3, and 4 (without repetition) that are divisible by 4 is **30**.