Find the no of ways arranging the letters of
the word SINGING so that
(i) They begin with and end with I
(ii) two G’s are come together
(iii) relative positions of vowels and
consonants are not disturbed

Given word SINGING
(i) The  number of words begin and end with I:-
I---------------I
The first and last place can be filled with I.
The Remaining 5 gaps can be filled with 5
letters in  $=\frac{5!}{2!2!}ways=\frac{120}{4}ways=30ways$
(ii) The two G’S  come together :-
Consider two G’s as 1 unit +5 letters are 5units
= 6 units
The 6 units are arranged in $\frac{6!}{2!2!}=\frac{720}{2\times 2}=180$ ways

The 2G’s are arranged themselves in $\frac{2!}{2!}=1$
required arrangements = 180 ways
(iii) relative positions of vowels and consonants
are not disturbed CVCCVCC
Two  vowels are arranged in 2 places $=\frac{2!}{2!}=1$
5 consonants are arranged in 5 places in
$=\frac{5!}{2!2!}=\frac{120}{4}=30$
total no of arrangements = $30\times 1=30$