How many 6 digit numbers can be formed out of the digits of the number 113113?

Options

A. a. 15
B. b. 180
C. c. 888
D. d. 222

Answer

To find how many 6-digit numbers can be formed from the digits of the number 113113, we can follow these steps:

### Step 1: Identify the digits and their frequencies
The number 113113 consists of the digits:
- 1 appears 4 times
- 3 appears 2 times

### Step 2: Use the formula for permutations of multiset
When calculating the number of distinct arrangements of a set of items where some items are identical, we use the formula:
\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2!} \]
where:
- \( n \) is the total number of items,
- \( n_1, n_2, \ldots \) are the frequencies of the identical items.

In our case:
- Total digits \( n = 6 \) (since we have 6 digits in total)
- \( n_1 = 4 \) (for the digit 1)
- \( n_2 = 2 \) (for the digit 3)

### Step 3: Plug in the values into the formula
Using the formula:
\[ \text{Number of arrangements} = \frac{6!}{4! \times 2!} \]

### Step 4: Calculate the factorials
Now, we calculate the factorials:
- \( 6! = 720 \)
- \( 4! = 24 \)
- \( 2! = 2 \)

### Step 5: Substitute the factorials into the formula
Now substituting the values:
\[ \text{Number of arrangements} = \frac{720}{24 \times 2} = \frac{720}{48} = 15 \]

### Step 6: Conclusion
Thus, the total number of distinct 6-digit numbers that can be formed from the digits of 113113 is **15**.

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