A five digits number divisible by 3 is to be formed using the number 0,1,2,3,4 and 5 without repetition. The number of such numbers are `m^(3)` then m is equal to

Explanation

To solve the problem of forming a five-digit number divisible by 3 using the digits 0, 1, 2, 3, 4, and 5 without repetition, we can follow these steps:

### Step 1: Understand the divisibility rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

### Step 2: Calculate the sum of the digits
The digits we have are 0, 1, 2, 3, 4, and 5. The sum of these digits is:
\[ 0 + 1 + 2 + 3 + 4 + 5 = 15 \]
Since 15 is divisible by 3, any combination of five digits from these six digits will also be divisible by 3 if the sum of the excluded digit is also divisible by 3.

### Step 3: Identify cases based on the excluded digit
We can exclude one digit at a time and check if the remaining digits' sum is divisible by 3.

1. **Exclude 0**: Remaining digits are 1, 2, 3, 4, 5.
- Sum = 15 (divisible by 3)
- Total arrangements = \(5! = 120\)

2. **Exclude 1**: Remaining digits are 0, 2, 3, 4, 5.
- Sum = 14 (not divisible by 3)
- Total arrangements = 0

3. **Exclude 2**: Remaining digits are 0, 1, 3, 4, 5.
- Sum = 13 (not divisible by 3)
- Total arrangements = 0

4. **Exclude 3**: Remaining digits are 0, 1, 2, 4, 5.
- Sum = 12 (divisible by 3)
- Total arrangements = \(4 \times 4! = 96\) (0 cannot be in the first position)

5. **Exclude 4**: Remaining digits are 0, 1, 2, 3, 5.
- Sum = 11 (not divisible by 3)
- Total arrangements = 0

6. **Exclude 5**: Remaining digits are 0, 1, 2, 3, 4.
- Sum = 10 (not divisible by 3)
- Total arrangements = 0

### Step 4: Calculate the total number of valid arrangements
From the above cases, we have:
- From excluding 0: 120 arrangements
- From excluding 3: 96 arrangements

Total arrangements = \(120 + 96 = 216\)

### Step 5: Express the total as \(m^3\)
We need to express 216 in the form \(m^3\):
\[ 216 = 6^3 \]
Thus, \(m = 6\).

### Final Answer
The value of \(m\) is \(6\).

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