Out of 20 consonants and 5 vowels, how many words can be formed containing 2 consonants and 2 vowels?

Answer

To solve the problem of how many words can be formed containing 2 consonants and 2 vowels from a set of 20 consonants and 5 vowels, we can break it down into a series of steps.

### Step-by-Step Solution:

1. **Select the Consonants**:
We need to choose 2 consonants from the 20 available consonants. The number of ways to choose 2 consonants from 20 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 20 \) and \( r = 2 \):
\[ \binom{20}{2} = \frac{20!}{2!(20-2)!} = \frac{20 \times 19}{2 \times 1} = 190 \]

2. **Select the Vowels**:
Next, we need to choose 2 vowels from the 5 available vowels. Again, we use the combination formula:
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

3. **Arrange the Selected Letters**:
After selecting 2 consonants and 2 vowels, we have a total of 4 letters to arrange. The number of ways to arrange 4 letters is given by \( 4! \):
\[ 4! = 24 \]

4. **Calculate the Total Number of Words**:
Now, we multiply the number of ways to choose the consonants, the number of ways to choose the vowels, and the number of arrangements:
\[ \text{Total Words} = \binom{20}{2} \times \binom{5}{2} \times 4! = 190 \times 10 \times 24 \]
Calculate this:
\[ 190 \times 10 = 1900 \]
\[ 1900 \times 24 = 45600 \]

Thus, the total number of words that can be formed is **45,600**.