Out of 7 consonants and 4 vowels, words are to be formed by involving 3 consonants and 2 vowels. The number of such words formed is `:`

Options

A. 25200
B. 22500
C. 10080
D. 5040

Answer

To solve the problem of forming words using 3 consonants and 2 vowels from a set of 7 consonants and 4 vowels, we can break the solution down into several steps.

### Step 1: Choose the consonants
We need to select 3 consonants from the 7 available consonants. This can be calculated using the combination formula:

\[ \text{Number of ways to choose 3 consonants from 7} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3! \cdot 4!} \]

Calculating this gives:

\[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

### Step 2: Choose the vowels
Next, we need to select 2 vowels from the 4 available vowels. Again, we use the combination formula:

\[ \text{Number of ways to choose 2 vowels from 4} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} \]

Calculating this gives:

\[ \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 3: Arrange the selected letters
After selecting 3 consonants and 2 vowels, we have a total of 5 letters (3 consonants + 2 vowels) to arrange. The number of ways to arrange these 5 letters is given by:

\[ \text{Number of arrangements of 5 letters} = 5! = 120 \]

### Step 4: Calculate the total number of words
Now, we can find the total number of words formed 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 selected letters:

\[ \text{Total number of words} = \binom{7}{3} \times \binom{4}{2} \times 5! = 35 \times 6 \times 120 \]

Calculating this gives:

\[ 35 \times 6 = 210 \]
\[ 210 \times 120 = 25200 \]

### Final Answer
Thus, the total number of words that can be formed is **25200**.

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