Find the number of permutations that can be made from the letters of the word 'OMEGA'. Vowels occupying odd places .

Options

A. a. 12 ways
B. b. 16 ways
C. c. 6 ways
D. d. 20 ways

Answer

To find the number of permutations of the letters in the word "OMEGA" with the condition that vowels occupy odd places, we can follow these steps:

### Step 1: Identify the letters and their types
The word "OMEGA" consists of 5 letters: O, M, E, G, A. Among these, the vowels are O, E, and A, and the consonants are M and G.

### Step 2: Determine the positions
In a 5-letter arrangement, the positions are numbered as follows:
1. Position 1 (odd)
2. Position 2 (even)
3. Position 3 (odd)
4. Position 4 (even)
5. Position 5 (odd)

The odd positions available are 1, 3, and 5.

### Step 3: Place the vowels in the odd positions
We have 3 vowels (O, E, A) that need to occupy the 3 odd positions (1, 3, and 5). The number of ways to arrange 3 vowels in 3 positions is calculated using the factorial of the number of vowels:
\[ 3! = 6 \]
So, there are 6 ways to arrange the vowels in the odd positions.

### Step 4: Place the consonants in the even positions
We have 2 consonants (M and G) that need to occupy the 2 even positions (2 and 4). The number of ways to arrange 2 consonants in 2 positions is:
\[ 2! = 2 \]
So, there are 2 ways to arrange the consonants in the even positions.

### Step 5: Calculate the total permutations
To find the total number of permutations, we multiply the number of arrangements of vowels by the number of arrangements of consonants:
\[ \text{Total permutations} = 3! \times 2! = 6 \times 2 = 12 \]

### Final Answer
Thus, the total number of permutations of the letters of the word "OMEGA" with vowels occupying odd places is **12**.

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