What is the total number of 4 digit numbers that can be formed using the digits 0,1,2,3,4,5 without repetition, such that the number is divisible by 9?

Options

A. 36
B. 28
C. 15
D. 18

Answer

To solve the problem of finding the total number of 4-digit numbers that can be formed using the digits 0, 1, 2, 3, 4, and 5 without repetition, such that the number is divisible by 9, we can follow these steps:

### Step 1: Identify the digits and their sum
The digits available are 0, 1, 2, 3, 4, and 5. The sum of these digits is:
\[ 0 + 1 + 2 + 3 + 4 + 5 = 15 \]
Since we need a 4-digit number, we will select 4 digits from these 6 digits.

### Step 2: Determine the condition for divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. Since the total sum of all digits (15) is not divisible by 9, we need to find combinations of 4 digits whose sum is divisible by 9.

### Step 3: Find valid combinations
We can find the combinations of 4 digits that can be formed from the available digits and check their sums:
1. **Combination 1:** 0, 1, 2, 3 → Sum = 6 (not divisible by 9)
2. **Combination 2:** 0, 1, 2, 4 → Sum = 7 (not divisible by 9)
3. **Combination 3:** 0, 1, 2, 5 → Sum = 8 (not divisible by 9)
4. **Combination 4:** 0, 1, 3, 4 → Sum = 8 (not divisible by 9)
5. **Combination 5:** 0, 1, 3, 5 → Sum = 9 (divisible by 9)
6. **Combination 6:** 0, 1, 4, 5 → Sum = 10 (not divisible by 9)
7. **Combination 7:** 0, 2, 3, 4 → Sum = 9 (divisible by 9)
8. **Combination 8:** 0, 2, 3, 5 → Sum = 10 (not divisible by 9)
9. **Combination 9:** 0, 2, 4, 5 → Sum = 11 (not divisible by 9)
10. **Combination 10:** 0, 3, 4, 5 → Sum = 12 (not divisible by 9)
11. **Combination 11:** 1, 2, 3, 4 → Sum = 10 (not divisible by 9)
12. **Combination 12:** 1, 2, 3, 5 → Sum = 11 (not divisible by 9)
13. **Combination 13:** 1, 2, 4, 5 → Sum = 12 (not divisible by 9)
14. **Combination 14:** 1, 3, 4, 5 → Sum = 13 (not divisible by 9)
15. **Combination 15:** 2, 3, 4, 5 → Sum = 14 (not divisible by 9)

The valid combinations are:
- **Combination 1:** 0, 1, 3, 5
- **Combination 2:** 0, 2, 3, 4

### Step 4: Calculate arrangements for each combination
For each combination, we need to calculate the number of valid 4-digit numbers.

1. **For the combination (0, 1, 3, 5):**
- Total arrangements = 4! = 24
- Invalid arrangements (where 0 is the first digit) = 3! = 6
- Valid arrangements = 24 - 6 = 18

2. **For the combination (0, 2, 3, 4):**
- Total arrangements = 4! = 24
- Invalid arrangements (where 0 is the first digit) = 3! = 6
- Valid arrangements = 24 - 6 = 18

### Step 5: Total valid arrangements
Now, we sum the valid arrangements from both combinations:
\[ Total = 18 + 18 = 36 \]

### Final Answer
Thus, the total number of 4-digit numbers that can be formed using the digits 0, 1, 2, 3, 4, 5 without repetition, such that the number is divisible by 9, is **36**.

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