Out of 7 consonants and 6 vowels ,how many words (with or without meaning ) can be made each containing 4 consonants and 2 vowels?

Options

A. `720xx7!`
B. `500xx7!`
C. `625xx6!`
D. `525xx6!`

Answer

To solve the problem of how many words can be formed using 4 consonants and 2 vowels from a set of 7 consonants and 6 vowels, we can break the solution down into several steps.

### Step-by-Step Solution:

1. **Identify the total consonants and vowels**:
- We have 7 consonants and 6 vowels.

2. **Choose the consonants**:
- We need to select 4 consonants from the 7 available consonants.
- The number of ways to choose 4 consonants from 7 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Therefore, the number of ways to choose 4 consonants from 7 is:
\[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \cdot 3!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

3. **Choose the vowels**:
- Next, we need to select 2 vowels from the 6 available vowels.
- The number of ways to choose 2 vowels from 6 is:
\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{6 \times 5}{2 \times 1} = 15 \]

4. **Calculate the total combinations**:
- Now, we multiply the number of ways to choose the consonants and the vowels:
\[ \text{Total combinations} = \binom{7}{4} \times \binom{6}{2} = 35 \times 15 = 525 \]

5. **Arrange the letters**:
- Each group of selected letters (4 consonants + 2 vowels = 6 letters) can be arranged in different ways.
- The number of ways to arrange 6 letters is given by \(6!\):
\[ 6! = 720 \]

6. **Calculate the total number of words**:
- Finally, we multiply the total combinations of letters by the arrangements of those letters:
\[ \text{Total words} = 525 \times 720 = 378000 \]

### Final Answer:
The total number of words that can be formed using 4 consonants and 2 vowels is **378000**.

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