Question

How many different words can be formed from the letters of the word GANESHPURI when:
The vowels always occupy even places (i.e. $2^{nd},4^{th}$ etc.)

Answer 1

GANESHPURI: No. of letters $=10$ vevels are AEUI i.e. 4 and consonants 6.

We have ten places out of which 5 places are old i.e. $1^{st},3^{rd},7^{th},9^{th}$ and five are even i.e. $2^{nd}$, fourth, sixth, eighth and tenth. In the five even places we have to fix up 4 vowels which can be done in ${}^5{P}_4$ ways. Having fixed up there vowels in even places, we will be left with six places namely 5 odd arid one even left after fixing the four vowels. In these six places we have to fix six consonants which can be done in ${}^6{P}_6$ i.e. $6!$ ways.

Thus the total number of ways is ${}^5{P}_4{\times }^6{P}_6$.

or $5!\times 6!=120\times 720=6(120{)}^2$.