How many 6 digit numbers can be formed with the digits 1,3,3,0,1,2?

Answer

To find how many 6-digit numbers can be formed using the digits 1, 3, 3, 0, 1, and 2, we need to follow these steps:

### Step-by-step Solution:

1. **Identify the Digits**: The digits we have are 1, 3, 3, 0, 1, and 2.

2. **Determine the First Digit**: Since we are forming a 6-digit number, the first digit cannot be 0. Therefore, the possible choices for the first digit are 1, 3, or 2.

3. **Case Analysis for the First Digit**:
- **Case 1**: First digit is 1.
- Remaining digits: 3, 3, 0, 1, 2.
- Total arrangements = \( \frac{5!}{2!} \) (since we have two 3's).
- **Case 2**: First digit is 3.
- Remaining digits: 1, 3, 0, 1, 2.
- Total arrangements = \( \frac{5!}{2!} \) (since we have two 1's).
- **Case 3**: First digit is 2.
- Remaining digits: 1, 3, 3, 0, 1.
- Total arrangements = \( \frac{5!}{2!} \) (since we have two 1's).

4. **Calculate the Total Arrangements for Each Case**:
- For each case, we calculate:
\[ 5! = 120 \]
\[ 2! = 2 \]
Therefore, for each case:
\[ \text{Total arrangements} = \frac{120}{2} = 60 \]

5. **Sum the Arrangements from All Cases**:
- Total arrangements = Case 1 + Case 2 + Case 3
- Total arrangements = 60 + 60 + 60 = 180

### Final Answer:
The total number of 6-digit numbers that can be formed with the digits 1, 3, 3, 0, 1, and 2 is **180**.