How many 6 digit numbers can be formed out of the digits of the number 113226 ?

Options

A. 156
B. 180
C. 280
D. 120

Answer

To solve the problem of how many 6-digit numbers can be formed from the digits of the number 113226, we can follow these steps:

### Step 1: Identify the digits and their frequencies
The digits in the number 113226 are:
- 1 (occurs 2 times)
- 2 (occurs 2 times)
- 3 (occurs 1 time)
- 6 (occurs 1 time)

### Step 2: Use the formula for permutations of multiset
The formula for the number of distinct permutations of a multiset is given by:

\[ \text{Number of permutations} = \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]

where:
- \( n \) is the total number of items,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the distinct items.

In our case:
- Total digits \( n = 6 \) (since we have 6 digits in total),
- Frequencies are \( n_1 = 2 \) (for digit 1), \( n_2 = 2 \) (for digit 2), \( n_3 = 1 \) (for digit 3), and \( n_4 = 1 \) (for digit 6).

### Step 3: Apply the values to the formula
Substituting the values into the formula, we have:

\[ \text{Number of permutations} = \frac{6!}{2! \times 2! \times 1! \times 1!} \]

### Step 4: Calculate the factorials
Now we calculate the factorials:
- \( 6! = 720 \)
- \( 2! = 2 \)
- \( 1! = 1 \)

So, we can substitute these values into our equation:

\[ \text{Number of permutations} = \frac{720}{2 \times 2 \times 1 \times 1} = \frac{720}{4} = 180 \]

### Step 5: Conclusion
Thus, the total number of distinct 6-digit numbers that can be formed from the digits of the number 113226 is **180**.

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