Question

Find the number of permutations that can be made from the letters of the word OMEGA, if the vowels occupy odd places.

• A. 12 ways
• B. 6 ways
• C. 13 ways
• D. 17 ways

Answer 1

The Correct Option is A

To solve the problem of finding the number of permutations of the letters in the word "OMEGA" with vowels occupying odd places, we first identify the vowels and consonants in the word. 

• The vowels in "OMEGA" are 'O', 'E', and 'A'.
• The consonants are 'M' and 'G'.

The positions need to be filled such that vowels are placed only in odd positions. The positions in a 5-letter word can be labeled as 1, 2, 3, 4, 5 where vowels should occupy odd positions (1, 3, 5).

Step 1: Placing vowels in odd positions

We have 3 odd positions (1, 3, and 5) and 3 vowels (O, E, A) to place in these positions.

The number of ways to arrange 3 vowels in the 3 odd positions is given by the factorial of 3:

\(3! = 3 \times 2 \times 1 = 6\)

Step 2: Placing consonants in even positions

There are 2 remaining positions (2, 4) which must be occupied by the 2 consonants 'M' and 'G'. The number of ways to arrange 2 consonants in these 2 positions is:

\(2! = 2 \times 1 = 2\)

Step 3: Total permutations with given constraints

Since these two actions (arranging vowels and arranging consonants) are independent, we can multiply the number of arrangements:

\(6 \times 2 = 12\)

Hence, the number of permutations of the letters of the word "OMEGA" with vowels occupying the odd places is 12 ways.

Conclusion: The correct answer is \(12\) ways.