Find the number of different words that can be formed from 15 consonants and 5 vowels by taking 2 consonants and 4 vowels in each word.

Options

A. 625
B. 378000
C. 525
D. 615

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: Determine the number of ways to choose consonants
We need to select 2 consonants from the 15 available consonants. The number of ways to choose 2 consonants from 15 is given by the combination formula:

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

### Step 2: Calculate \(\binom{15}{2}\)
Using the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

For our case:

\[ \binom{15}{2} = \frac{15!}{2!(15-2)!} = \frac{15!}{2! \cdot 13!} \]

This simplifies to:

\[ \binom{15}{2} = \frac{15 \times 14}{2 \times 1} = \frac{210}{2} = 105 \]

### Step 3: Determine the number of ways to choose vowels
Next, we need to select 4 vowels from the 5 available vowels. The number of ways to choose 4 vowels from 5 is given by:

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

### Step 4: Calculate \(\binom{5}{4}\)
Using the combination formula again:

\[ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \cdot 1!} \]

This simplifies to:

\[ \binom{5}{4} = \frac{5}{1} = 5 \]

### Step 5: Calculate the total number of combinations
Now, we multiply the number of ways to choose the consonants by the number of ways to choose the vowels:

\[ \text{Total combinations} = \binom{15}{2} \times \binom{5}{4} = 105 \times 5 = 525 \]

### Step 6: Conclusion
Thus, the total number of different words that can be formed by taking 2 consonants and 4 vowels is:

\[ \text{Total different words} = 525 \]

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