Question

How many different words can be formed from the letter of the word 'GANESHPURI'? In how many of these words:
(i)the letter G always occupies the first place?
(ii) the letter P and I respectively occupy first and last place?
(iii) the vowels are always together?

Answer 1

Number of letters$=10$

Vowels are A,E,I,U i.e. 4 vowels and 6 consonants

(i) If we fixed $G$ at first place then we are left with the permutation of remaining $9$ letters which can be arranged in $9!$ ways

(ii) If $P$ occupies first and $I$ occupies the last place so we have to arrange the remaining $8$ letters which can be done in $8!$ ways.

(iii) The four vowels $AEIU$ are to be together. Take these four letters as one letter and so we in all $10-4+1=7$ letters which can be arranged in $7!$ ways. In each of these $7!$ arrangements of the four vowels are together and these can be among themselves in $4!$ ways. Hence by fundamental theorem the number of ways will be $7!\cdot 4!$ in which all vowels will be together.