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

Answer

To solve the problem of how many 6-digit numbers can be formed using the digits 1, 3, 3, 0, 1, 2, we will follow these steps:

### Step 1: Identify the digits and restrictions
We have the digits: 1, 3, 3, 0, 1, 2. Since we are forming a 6-digit number, the first digit cannot be 0 (as it would not be a valid 6-digit number).

### Step 2: Determine valid choices for the first digit
The valid choices for the first digit are 1, 2, or 3. We will consider each case separately.

### Step 3: Case 1 - First digit is 1
If the first digit is 1, the remaining digits are 3, 3, 0, 1, 2. We have the digits: 3, 3, 0, 1, 2.

- Total digits left = 5 (3, 3, 0, 1, 2)
- The number of arrangements of these digits, accounting for the repetition of 3's and 1's, is given by:

\[ \frac{5!}{2! \cdot 1!} = \frac{120}{2} = 60 \]

### Step 4: Case 2 - First digit is 2
If the first digit is 2, the remaining digits are 1, 3, 3, 0, 1. We have the digits: 1, 3, 3, 0, 1.

- Total digits left = 5 (1, 3, 3, 0, 1)
- The number of arrangements of these digits, accounting for the repetition of 3's and 1's, is given by:

\[ \frac{5!}{2! \cdot 2!} = \frac{120}{4} = 30 \]

### Step 5: Case 3 - First digit is 3
If the first digit is 3, the remaining digits are 1, 3, 0, 1, 2. We have the digits: 1, 3, 0, 1, 2.

- Total digits left = 5 (1, 3, 0, 1, 2)
- The number of arrangements of these digits, accounting for the repetition of 1's, is given by:

\[ \frac{5!}{2!} = \frac{120}{2} = 60 \]

### Step 6: Calculate the total number of valid arrangements
Now, we sum the valid arrangements from all three cases:

- Case 1 (first digit 1): 60
- Case 2 (first digit 2): 30
- Case 3 (first digit 3): 60

Total = 60 + 30 + 60 = 150

### Final Answer
Thus, the total number of 6-digit numbers that can be formed with the digits 1, 3, 3, 0, 1, 2 is **150**.

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