How many 7 digit numbers can be formed with the digits 2,3,0,3,4,4,3?

Answer

To solve the problem of how many 7-digit numbers can be formed with the digits 2, 3, 0, 3, 4, 4, 3, we need to follow these steps:

### Step 1: Identify the digits and their frequencies
The digits we have are:
- 2 (1 time)
- 3 (3 times)
- 0 (1 time)
- 4 (2 times)

### Step 2: Determine the restrictions
Since we are forming a 7-digit number, the first digit cannot be 0. Therefore, we can only use the digits 2, 3, or 4 as the first digit.

### Step 3: Calculate the total arrangements for each case
We will consider three cases based on the choice of the first digit:

#### Case 1: First digit is 2
The remaining digits are 3, 3, 3, 0, 4, 4.
- Total arrangements = \(\frac{6!}{3! \cdot 2! \cdot 1!}\)

Calculating this:
\[ 6! = 720 \]
\[ 3! = 6, \quad 2! = 2, \quad 1! = 1 \]
\[ \text{Total arrangements} = \frac{720}{6 \cdot 2 \cdot 1} = \frac{720}{12} = 60 \]

#### Case 2: First digit is 3
The remaining digits are 2, 3, 3, 0, 4, 4.
- Total arrangements = \(\frac{6!}{2! \cdot 2! \cdot 1! \cdot 1!}\)

Calculating this:
\[ \text{Total arrangements} = \frac{720}{2 \cdot 2 \cdot 1 \cdot 1} = \frac{720}{4} = 180 \]

#### Case 3: First digit is 4
The remaining digits are 2, 3, 3, 0, 3, 4.
- Total arrangements = \(\frac{6!}{3! \cdot 1! \cdot 1! \cdot 1!}\)

Calculating this:
\[ \text{Total arrangements} = \frac{720}{6 \cdot 1 \cdot 1 \cdot 1} = \frac{720}{6} = 120 \]

### Step 4: Add the arrangements from all cases
Now, we sum the total arrangements from all three cases:
\[ \text{Total} = 60 + 180 + 120 = 360 \]

### Final Answer
The total number of 7-digit numbers that can be formed with the digits 2, 3, 0, 3, 4, 4, 3 is **360**.

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