Given : the word is ‘VOWELS.’
To find : number of words in which consonants always come together
Number of consonants in this word = 4(V, W, L, S)
Now,
Consider these four consonants as one entity(VWLS together as a single letter)
So,
The total number of letters = 3 (VWLS O E)
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 3 things taken all at a time
= P(3, 3)
= \(\frac{3!}{(3-3)!}\)
= \(\frac{3!}{0!}\)
{∵ 0! = 1}
= 3!
= 3 × 2 × 1
= 6
Now,
4 consonants which are together as a letter can be arranged in 4! (like WLVS or SWLV)
= 4 × 3 × 2 × 1
= 24 ways
Total number of words in which vowels come together = 24 × 6
= 144
