Question

Letters of the word "REVOLUTION" are to be
arranged. In how many ways can this be done
so that (i) No two vowels are together (ii) Not
all vowels are together?​

Answer 1

Answer:

(i). Number of ways such that all vowels are together is 86400.

(ii). Number of ways such that no vowels are together is 3542400.

Step-by-step explanation:

Given Word,

" REVOLUTION"

To find: (i) All vowels together

             (ii) No Two Vowels together.

Total alphabet in word = 10

Number of vowels in word = 5

Number of consonant in word = 5

(i).

Consider all vowels a single unit.

Then we have to place 5 consonant and 1 unit of vowels in 6 places.

Number of ways of this arrangement = $^6P_6$ = 6! = 720

Vowels can also be changed among them. So, Number of way vowels arranged = 5! = 120

Total Number of ways such that all vowels are together = 720 × 120 = 86400

(ii).

Total Number ways all alphabets can be arranged = 10! = 3628800

So, Number of ways such that no vowels are together = 3628800 - 86400 = 3542400