Out of 12 consonants and 5 vowels, how many words of 2 consonants and 3 vowels can be formed?

Answer

To find the total number of words that can be formed using 2 consonants and 3 vowels from a set of 12 consonants and 5 vowels, we can follow these steps:

### Step 1: Choose 2 consonants from 12 consonants
We can choose 2 consonants from 12 using the combination formula:
\[ \text{Number of ways to choose 2 consonants} = \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]

### Step 2: Choose 3 vowels from 5 vowels
Next, we choose 3 vowels from 5 using the combination formula:
\[ \text{Number of ways to choose 3 vowels} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Calculate the total number of letters
After choosing 2 consonants and 3 vowels, we have a total of:
\[ 2 + 3 = 5 \text{ letters} \]

### Step 4: Find the number of permutations of the 5 letters
The number of different arrangements (permutations) of these 5 letters is given by:
\[ 5! = 120 \]

### Step 5: Calculate the total number of words
Now, we can find the total number of words by multiplying the number of ways to choose the consonants, the number of ways to choose the vowels, and the number of permutations of the letters:
\[ \text{Total number of words} = \binom{12}{2} \times \binom{5}{3} \times 5! = 66 \times 10 \times 120 \]

Calculating this gives:
\[ 66 \times 10 = 660 \]
\[ 660 \times 120 = 79200 \]

Thus, the total number of words that can be formed is **79200**.

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