With 17 consonants and 5 vowels the number of words of four letters that can be formed having two different vowels in the middle and one consonant, repeated or different at each end is

Options

A. 5780
B. 2890
C. 5440
D. 2720

Answer

To solve the problem of forming four-letter words with specific conditions, we will follow these steps:

### Step-by-Step Solution

1. **Identify the structure of the word**:
The word consists of four letters in the format: C V V C, where C represents consonants and V represents vowels. The two middle letters must be different vowels.

2. **Select the vowels**:
We have 5 vowels to choose from, and we need to select 2 different vowels for the middle positions. The number of ways to choose 2 different vowels from 5 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Ways to choose 2 vowels} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

3. **Arrange the vowels**:
The two vowels selected can be arranged in the two middle positions in \( 2! \) ways:
\[ \text{Ways to arrange 2 vowels} = 2! = 2 \]
Therefore, the total ways to fill the middle positions with vowels is:
\[ \text{Total ways for vowels} = 10 \times 2 = 20 \]

4. **Select the consonants**:
For the first and last positions, we can use any of the 17 consonants. Since the consonants can be the same or different, we have:
\[ \text{Ways to choose the first consonant} = 17 \quad \text{(any of the 17 consonants)} \]
\[ \text{Ways to choose the last consonant} = 17 \quad \text{(any of the 17 consonants)} \]
Thus, the total ways to fill the consonant positions is:
\[ \text{Total ways for consonants} = 17 \times 17 = 289 \]

5. **Combine the choices**:
The total number of four-letter words that can be formed is the product of the ways to choose and arrange the vowels and the ways to choose the consonants:
\[ \text{Total number of words} = \text{Total ways for vowels} \times \text{Total ways for consonants} \]
\[ \text{Total number of words} = 20 \times 289 = 5780 \]

### Final Answer
The total number of four-letter words that can be formed is **5780**.