How many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4 , 2, 3?

Options

A. 360
B. 240
C. 480
D. 460

Answer

To find how many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4, 2, 3, we can follow these steps:

### Step 1: Understand the requirement
A number greater than a million must have 7 digits. The digits we have are 2, 3, 0, 3, 4, 2, 3.

### Step 2: Identify the first digit
The first digit cannot be 0, as that would make it a 6-digit number. The possible digits for the first position are 2, 3, or 4.

### Step 3: Count the choices for the first digit
- If the first digit is 2, the remaining digits are 3, 0, 3, 4, 2, 3.
- If the first digit is 3, the remaining digits are 2, 0, 3, 4, 2, 3.
- If the first digit is 4, the remaining digits are 2, 3, 0, 3, 2, 3.

### Step 4: Calculate the total arrangements for each case
For each case, we will calculate the number of arrangements of the remaining 6 digits, taking into account the repetitions of digits.

#### Case 1: First digit is 2
Remaining digits: 3, 0, 3, 4, 2, 3
- Total arrangements = 6! / (3!) = 720 / 6 = 120

#### Case 2: First digit is 3
Remaining digits: 2, 0, 3, 4, 2, 3
- Total arrangements = 6! / (2!) = 720 / 2 = 360

#### Case 3: First digit is 4
Remaining digits: 2, 3, 0, 3, 2, 3
- Total arrangements = 6! / (2!) = 720 / 2 = 360

### Step 5: Sum the arrangements
Now, we add the number of arrangements from all three cases:
- Case 1: 120
- Case 2: 360
- Case 3: 360

Total = 120 + 360 + 360 = 840

### Final Answer
The total number of numbers greater than a million that can be formed with the digits 2, 3, 0, 3, 4, 2, 3 is **840**.

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