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