How many permutations can be formed by the letters of the word, ‘VOWELS,’ when
all consonants come together?
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)
∴ Total number of arrangements
= the number of arrangements of 3 things taken all at a time
= P(3, 3)
{∵ 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