How many 6 digit numbers can be formed using the digit 2 two times and the digit 4 four times?

Options

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

Answer

To find how many 6-digit numbers can be formed using the digit 2 two times and the digit 4 four times, we can follow these steps:

### Step 1: Identify the total number of digits and their frequencies
We have a total of 6 digits to arrange:
- The digit 2 appears 2 times.
- The digit 4 appears 4 times.

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

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

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

In our case:
- \( n = 6 \) (total digits),
- \( n_1 = 2 \) (for digit 2),
- \( n_2 = 4 \) (for digit 4).

### Step 3: Substitute the values into the formula
Now we can substitute the values into the formula:

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

### 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 substitute the factorial values back into the equation:

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

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

\[ 2 \times 24 = 48 \]

Now divide:

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

### Final Answer
Thus, the total number of 6-digit numbers that can be formed is **15**.

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