Solution:
The word ‘STRANGE’ has seven letters, including two vowels (A,E) and five consonants (S,T,R,N,G).
(i) the vowels come together?
If we consider two vowels to be one letter, we’ll end up with six letters that can be ordered in six different ways.
(A,E) can be combined in 2P2 ways.
As a result, the needed word count is
Using the formula, we can
$ P\text{ }\left( n,\text{ }r \right)\text{ }=\text{ }n!/\left( n-r \right)! $
$ P\text{ }\left( 6,\text{ }6 \right)\text{ }\times \text{ }P\text{ }\left( 2,\text{ }2 \right)\text{ }=\text{ }6!/\left( 6-6 \right)!\text{ }\times \text{ }2!/\left( 2-2 \right)! $
$ =\text{ }6!\text{ }\times \text{ }2! $
$ =\text{ }6\text{ }\times \text{ }5\text{ }\times \text{ }4\text{ }\times \text{ }3\text{ }\times \text{ }2\text{ }\times \text{ }1\text{ }\times \text{ }2\text{ }\times \text{ }1 $
$ =\text{ }720\text{ }\times \text{ }2 $
$ =\text{ }1440 $
(ii) the vowels never come together?
The total number of letters in the word ‘STRANGE’ is given by:
7P7 = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
7P7 = 5040
So,
Total number of words where vowels aren’t together = total number of words – total number of words in which vowels are always together
= 5040 – 1440 = 3600
As a result, there are 3600 configurations in which vowels never come together.