How many number greater than 3,00,000 can be formed by using all the digits of the number 111223?

Answer

To solve the problem of how many numbers greater than 3,00,000 can be formed using all the digits of the number 111223, we can follow these steps:

### Step 1: Identify the total digits and their counts
The number 111223 consists of the digits:
- 1 appears 3 times
- 2 appears 2 times
- 3 appears 1 time

### Step 2: Determine the condition for the number to be greater than 3,00,000
For a number to be greater than 3,00,000, the first digit must be either 1, 2, or 3. However, since we want numbers greater than 3,00,000, the first digit must be 3.

### Step 3: Fix the first digit as 3
By fixing the first digit as 3, we have the following digits left to arrange:
- 1, 1, 1, 2, 2 (total of 5 digits)

### Step 4: Calculate the arrangements of the remaining digits
To find the number of distinct arrangements of the digits 1, 1, 1, 2, 2, we use the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times \ldots} \]
Where:
- \( n \) is the total number of items to arrange (5 in this case),
- \( n_1, n_2, \ldots \) are the counts of each distinct item.

Here, we have:
- Total digits = 5 (1, 1, 1, 2, 2)
- Count of 1's = 3
- Count of 2's = 2

Thus, the number of arrangements is:
\[ \text{Number of arrangements} = \frac{5!}{3! \times 2!} \]

### Step 5: Calculate the factorials
Calculating the factorials:
- \( 5! = 120 \)
- \( 3! = 6 \)
- \( 2! = 2 \)

### Step 6: Substitute the values into the formula
Now substituting the values into the formula:
\[ \text{Number of arrangements} = \frac{120}{6 \times 2} = \frac{120}{12} = 10 \]

### Conclusion
Thus, the total number of distinct numbers greater than 3,00,000 that can be formed using the digits of 111223 is **10**.

---