Out of 7 consonants and 4 vowels, how many words can be made each containing 3 consonants and 2 vowels ?

Answer

To solve the problem of how many words can be formed using 3 consonants and 2 vowels from a set of 7 consonants and 4 vowels, we can follow these steps:

### Step 1: Choose the consonants
We need to select 3 consonants from the 7 available consonants. The number of ways to choose 3 consonants from 7 can be calculated using the combination formula:

\[ \text{Number of ways to choose 3 consonants} = \binom{7}{3} = \frac{7!}{(7-3)! \cdot 3!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

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

\[ \text{Number of ways to choose 2 vowels} = \binom{4}{2} = \frac{4!}{(4-2)! \cdot 2!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 3: Arrange the chosen letters
Now that we have chosen 3 consonants and 2 vowels, we need to arrange these 5 letters. The number of ways to arrange 5 letters is given by the factorial of the number of letters:

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

### Step 4: Calculate the total number of words
To find the total number of words that can be formed, we multiply the number of ways to choose the consonants, the number of ways to choose the vowels, and the number of arrangements:

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

Calculating this step-by-step:

1. \( 35 \times 6 = 210 \)
2. \( 210 \times 120 = 25200 \)

Thus, the total number of words that can be formed is **25,200**.

### Summary of Steps:
1. Choose 3 consonants from 7: \( \binom{7}{3} = 35 \)
2. Choose 2 vowels from 4: \( \binom{4}{2} = 6 \)
3. Arrange 5 letters: \( 5! = 120 \)
4. Total words: \( 35 \times 6 \times 120 = 25200 \)