Number of 6 digit numbers that can be formed using digit 2 two times and the digit 5 four times is

Options

A. 16
B. 15
C. 24
D. 18

Answer

To find the number of 6-digit numbers that can be formed using the digit 2 two times and the digit 5 four times, we can follow these steps:

### Step 1: Identify the total number of digits
We have a total of 6 digits, which consists of:
- The digit 2 appearing 2 times
- The digit 5 appearing 4 times

### Step 2: Use the formula for permutations of multiset
The formula to calculate the number of arrangements of n items where there are groups of indistinguishable items is given by:

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2!} \]

Where:
- \( n \) is the total number of items
- \( n_1, n_2, \ldots \) are the counts of each indistinguishable item

### Step 3: Substitute the values into the formula
In our case:
- Total digits \( n = 6 \)
- The digit 2 appears \( n_1 = 2 \) times
- The digit 5 appears \( n_2 = 4 \) times

So, we can substitute these values into the formula:

\[ \text{Number of arrangements} = \frac{6!}{2! \times 4!} \]

### Step 4: Calculate the factorials
Now, we calculate the factorials:

- \( 6! = 720 \)
- \( 2! = 2 \)
- \( 4! = 24 \)

### Step 5: Substitute the factorial values into the formula
Now we can substitute these values back into the equation:

\[ \text{Number of arrangements} = \frac{720}{2 \times 24} \]

### Step 6: Simplify the expression
Calculating the denominator:

\[ 2 \times 24 = 48 \]

Now divide:

\[ \text{Number of arrangements} = \frac{720}{48} = 15 \]

### Conclusion
Thus, the total number of 6-digit numbers that can be formed using the digit 2 two times and the digit 5 four times is **15**.