1. Find the number of permutations that can be had from the letters of the word 'OMEGA' such that

i) O & Aoccupy odd places
ii) E always occupies the middle place
iii) Vowels occupy odd places
iv) All the vowels are never together

Dear Student Your friend has provided the correct answer.

@Harmeet: Good effort! Keep posting!

For part(1):

Second step: O and A are two letters to be arranged = 3p2.2!

(Explanation: Two letters O and A can be arranged in three odd place in 3p2 ways.

Now, these two letters can be arranged among themselves in 2! ways.

So, no. of ways to arrange O and A = 3p2.2!)

Rest everything is same.

Got it!

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    there are three odd places i.e=1,3,5

   A and O are two letters to be arranged=3C2.2!

   the other three letters can be arranged in 3! ways

so total ways=6X6=36 ways

2. Fix the E in the middle place the other four letters can be arranged in 4! ways=24

3.There are 3 vowels and three odd places so vowels can be arranged in 3! ways

    the other 2 letters can be arranged in 2! ways

   so total no. of ways=6X2=12

4.for the two consonents there are 2! ways and vowels can be arranged between them in 3! ways

  total no. of ways = 2X6=12

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