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How many words can be formed with the letters of the word ‘OMEGA’ when vowels are never together?
(a) 12
(b) 36
(c) 24
(d) 84

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Last updated date: 12th Jul 2024
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Answer
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Hint: We have given the word OMEGA and condition that when vowels are never together. In this problem we will use the combination and permutations for finding our value which is used for that. First, we will calculate the number of ways for vowels together and then we find the total number of ways. And then by subtracting from each other we will finally get our answer.

Complete step by step answer:
According to the question a word ‘OMEGA’ is given and with the condition ‘vowels being never together’ it is asked to form other words from these letters of OMEGA.
If we understand the combination by an example then it will be clear.
For example, if we want to buy a milkshake and we are allowed to combine three flavours from Apple, Banana, cherry and papaya. Then the combination of Apple, Banana and cherry is the same as the combination of Banana, Apple and cherry. So, if we are supposed to make a combination out of these possible flavours, then firstly, we will shorten the name of the fruits by selecting the first letter of their name. And then the combination of four is ABC, ABP, ACP, BCP.
So, the total number of ways are 4 here.
And we can also calculate this by formula
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\]
Here, the word given to us is OMEGA.
But a condition is also given to us that no vowels will be together here.
So, by using this condition we will find a number of ways when vowels are not together.
So, number of way when vowels are not together \[=3!\times 3!=36\]
Now, we find the total number of ways for arranging letters of omega \[=5!=120\]
Now, words when vowels will not be together \[=120-36=84\]
So, the total number of ways according to condition are 84.

So, the correct answer is “Option d”.

Note: While solving this type of question you have to have a good knowledge of vowels and consonants and you must apply the condition given in question. If that condition will leave then the total result will be wrong and all is wasted. So, read carefully to question and condition mainly.