Out of 10 consonants and four vowels, the number of words that can be formed using six consonants and three vowels is

Options

A. `""^(10)P_(6) xx ""^(6)P_(3)`
B. `""^(10)C_(6) xx ""^(6)C_(3)`
C. `""^(10)C_(6) xx ""^(4)C_(3) xx 9!`
D. `""^(10) P_(6) xx ""^(4)P_(3)`

Answer

To solve the problem of finding the number of words that can be formed using six consonants and three vowels from a set of 10 consonants and 4 vowels, we can follow these steps:

### Step 1: Choose the consonants
We need to choose 6 consonants from a total of 10 consonants. The number of ways to choose 6 consonants from 10 can be calculated using the combination formula:

\[ \text{Number of ways to choose consonants} = \binom{10}{6} \]

### Step 2: Choose the vowels
Next, we need to choose 3 vowels from a total of 4 vowels. The number of ways to choose 3 vowels from 4 can also be calculated using the combination formula:

\[ \text{Number of ways to choose vowels} = \binom{4}{3} \]

### Step 3: Arrange the chosen letters
After selecting 6 consonants and 3 vowels, we have a total of 9 letters (6 consonants + 3 vowels). The number of ways to arrange these 9 letters can be calculated using the factorial of the total number of letters:

\[ \text{Number of arrangements} = 9! \]

### Step 4: Combine the results
Now, we can combine the results from the previous steps to find the total number of words that can be formed:

\[ \text{Total number of words} = \binom{10}{6} \times \binom{4}{3} \times 9! \]

### Step 5: Calculate the values
Now we can calculate the values of the combinations and the factorial:

1. Calculate \(\binom{10}{6}\):
\[ \binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!} = 210 \]

2. Calculate \(\binom{4}{3}\):
\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = 4 \]

3. Calculate \(9!\):
\[ 9! = 362880 \]

### Step 6: Final Calculation
Now, substituting these values into the total number of words formula:

\[ \text{Total number of words} = 210 \times 4 \times 362880 \]

Calculating this gives:

\[ \text{Total number of words} = 210 \times 4 = 840 \]
\[ 840 \times 362880 = 304819200 \]

Thus, the total number of words that can be formed is **304,819,200**.