The number of 6-digit numbers that can be formed using the three digits 0,1 and 2, is

Options

A. `3^(6)`
B. `2xx3^(5)`
C. `3^(5)`
D. none of these

Explanation

To find the number of 6-digit numbers that can be formed using the digits 0, 1, and 2, we need to follow these steps:

### Step 1: Identify the conditions for a 6-digit number
A 6-digit number cannot start with the digit 0, as this would make it a 5-digit number. Therefore, the first digit must be either 1 or 2.

### Step 2: Determine choices for the first digit
Since the first digit can only be 1 or 2, we have 2 choices for the first digit.

### Step 3: Determine choices for the remaining digits
For the remaining 5 digits, we can use any of the three digits: 0, 1, or 2. Therefore, for each of the remaining 5 positions, we have 3 choices.

### Step 4: Calculate the total number of combinations
The total number of 6-digit numbers can be calculated as follows:
- Choices for the first digit: 2
- Choices for the remaining 5 digits: \(3^5\)

Thus, the total number of 6-digit numbers is:
\[ \text{Total} = 2 \times 3^5 \]

### Step 5: Calculate \(3^5\)
Calculating \(3^5\):
\[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \]

### Step 6: Multiply to find the final answer
Now, we multiply the choices for the first digit by the choices for the remaining digits:
\[ \text{Total} = 2 \times 243 = 486 \]

### Conclusion
The total number of 6-digit numbers that can be formed using the digits 0, 1, and 2 is **486**.

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