Number of different words that can be formed from 15 consonants and 5 vowels by taking 2 consonants and 4 vowels in each word is

Options

A. `""^(15)P_(2) xx ""^(5)C_(4)`
B. `""^(15)C_(2)xx""^(5)C_(4)xx6!`
C. `""^(15)P_(2)`
D. `""^(15)P_(2)xx""^(5)P_(4)xx2!xx4!`

Answer

To solve the problem of finding the number of different words that can be formed from 15 consonants and 5 vowels by taking 2 consonants and 4 vowels, we can follow these steps:

### Step 1: Calculate the number of ways to choose consonants
We need to select 2 consonants from a total of 15 consonants. The number of ways to choose 2 consonants from 15 can be calculated using the combination formula:

\[ \text{Number of ways to choose 2 consonants} = \binom{15}{2} \]

### Step 2: Calculate the number of ways to choose vowels
Next, we need to select 4 vowels from a total of 5 vowels. The number of ways to choose 4 vowels from 5 can also be calculated using the combination formula:

\[ \text{Number of ways to choose 4 vowels} = \binom{5}{4} \]

### Step 3: Calculate the total number of characters
After selecting 2 consonants and 4 vowels, we will have a total of:

\[ 2 \text{ (consonants)} + 4 \text{ (vowels)} = 6 \text{ characters} \]

### Step 4: Calculate the number of arrangements of the characters
The total number of ways to arrange these 6 characters (2 consonants and 4 vowels) is given by the factorial of the number of characters:

\[ \text{Number of arrangements} = 6! \]

### Step 5: Calculate the total number of different words
Now, we can find the total number of different words by multiplying the number of ways to choose the consonants, the number of ways to choose the vowels, and the number of arrangements of the characters:

\[ \text{Total number of words} = \binom{15}{2} \times \binom{5}{4} \times 6! \]

### Step 6: Calculate the values
Now, let's compute the values:

1. \(\binom{15}{2} = \frac{15 \times 14}{2 \times 1} = 105\)
2. \(\binom{5}{4} = 5\)
3. \(6! = 720\)

Now, substituting these values back into the total number of words formula:

\[ \text{Total number of words} = 105 \times 5 \times 720 \]

### Step 7: Final Calculation
Calculating this gives:

\[ 105 \times 5 = 525 \]
\[ 525 \times 720 = 378000 \]

Thus, the total number of different words that can be formed is **378000**.