Number of words each consisting of two vowels and two consonants which can be make out of the latters of the word DEVASTATION is 126 (b) 198 (c) 1512 (d) 1638

Options

A. 126
B. 198
C. 1512
D. 1638

Answer

To solve the problem of finding the number of words consisting of two vowels and two consonants that can be formed from the letters of the word "DEVASTATION," we will follow these steps:

### Step 1: Identify the vowels and consonants
The letters in "DEVASTATION" are:
- Vowels: A, A, E, I, O (4 vowels, with 'A' repeating)
- Consonants: D, V, S, T, T, N (5 consonants, with 'T' repeating)

### Step 2: Calculate the number of combinations for different cases
We will consider four cases based on the repetition of vowels and consonants.

#### Case 1: Both vowels and consonants are different
- Choose 2 different vowels from 4: \( \binom{4}{2} \)
- Choose 2 different consonants from 5: \( \binom{5}{2} \)
- Arrange the 4 letters (2 vowels + 2 consonants): \( 4! \)

Calculating:
\[ \text{Ways} = \binom{4}{2} \times \binom{5}{2} \times 4! = 6 \times 10 \times 24 = 1440 \]

#### Case 2: Vowels are the same, consonants are different
- Choose 2 identical vowels (A, A): \( \binom{2}{2} = 1 \)
- Choose 2 different consonants from 5: \( \binom{5}{2} \)
- Arrange the letters (2 A's + 2 different consonants): \( \frac{4!}{2!} \)

Calculating:
\[ \text{Ways} = 1 \times \binom{5}{2} \times \frac{4!}{2!} = 1 \times 10 \times 12 = 120 \]

#### Case 3: Vowels are different, consonants are the same
- Choose 2 different vowels from 4: \( \binom{4}{2} \)
- Choose 2 identical consonants (T, T): \( \binom{2}{2} = 1 \)
- Arrange the letters (2 different vowels + 2 T's): \( \frac{4!}{2!} \)

Calculating:
\[ \text{Ways} = \binom{4}{2} \times 1 \times \frac{4!}{2!} = 6 \times 1 \times 12 = 72 \]

#### Case 4: Both vowels and consonants are the same
- Choose 2 identical vowels (A, A): \( \binom{2}{2} = 1 \)
- Choose 2 identical consonants (T, T): \( \binom{2}{2} = 1 \)
- Arrange the letters (2 A's + 2 T's): \( \frac{4!}{2! \times 2!} \)

Calculating:
\[ \text{Ways} = 1 \times 1 \times \frac{4!}{2! \times 2!} = 1 \times 1 \times 6 = 6 \]

### Step 3: Sum all cases
Now, we will sum the number of ways from all four cases:
\[ \text{Total Ways} = 1440 + 120 + 72 + 6 = 1638 \]

### Final Answer
Thus, the total number of words consisting of two vowels and two consonants that can be formed from the letters of "DEVASTATION" is **1638**.