Find the number of different words that can be formed from 12 consonants and 5 vowels by taking 4 consonants and 3 vowels in each words.

Answer

To solve the problem of finding the number of different words that can be formed from 12 consonants and 5 vowels by taking 4 consonants and 3 vowels, we can break down the solution into a series of steps:

### Step 1: Select the Consonants
We need to select 4 consonants from a total of 12 consonants. The number of ways to choose 4 consonants from 12 can be calculated using the combination formula:

\[ \text{Number of ways to choose consonants} = \binom{12}{4} \]

### Step 2: Select the Vowels
Next, we need to select 3 vowels from a total of 5 vowels. The number of ways to choose 3 vowels from 5 can also be calculated using the combination formula:

\[ \text{Number of ways to choose vowels} = \binom{5}{3} \]

### Step 3: Calculate the Total Combinations
Now, we can calculate the total number of ways to select the consonants and vowels:

\[ \text{Total combinations} = \binom{12}{4} \times \binom{5}{3} \]

### Step 4: Arrange the Selected Letters
After selecting 4 consonants and 3 vowels, we have a total of 7 letters (4 consonants + 3 vowels). The number of ways to arrange these 7 letters is given by:

\[ \text{Number of arrangements} = 7! \]

### Step 5: Combine the Results
Finally, the total number of different words that can be formed is the product of the combinations and the arrangements:

\[ \text{Total words} = \binom{12}{4} \times \binom{5}{3} \times 7! \]

### Step 6: Calculate the Values
Now, we will calculate each of these values:

1. Calculate \(\binom{12}{4}\):

\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

2. Calculate \(\binom{5}{3}\):

\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

3. Calculate \(7!\):

\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

### Step 7: Final Calculation
Now, we can substitute these values back into our total words formula:

\[ \text{Total words} = 495 \times 10 \times 5040 \]

Calculating this gives:

\[ \text{Total words} = 4950 \times 5040 = 24948000 \]

Thus, the total number of different words that can be formed is **24,948,000**.