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How many permutations can be formed by the letters of the word, ‘VOWELS,’ when all vowels come together?

1 Answer

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Best answer

Given : the word is ‘VOWELS.’ 

To find : number of words in which vowels always come together 

Number of vowels in this word = 2(O, E) 

Now,

Consider these two vowels as one entity(OE together as a single letter) 

So, the total number of letters = 5 (OE V W L S) 

Formula used :

Number of arrangements of n things taken all at a time = P(n, n)

P(n, r) = \(\frac{n!}{(n-r)!}\)

∴ Total number of arrangements 

= the number of arrangements of 5 things taken all at a time 

= P(5, 5)

\(\frac{5!}{(5-5)!}\)

 = \(\frac{5!}{0!}\)

= 5! 

= 5 × 4 × 3 × 2 × 1 

= 120 

Now,

2 vowels which are together as a letter can be arranged in 2! (like OE or EO) 

= 2 × 1 

= 2 ways 

Total number of words in which vowels come together = 2 × 120

= 240

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