Question

How many different words can be formed from the letters of the word GANESHPURI when:
How many words of 5 letters each can be formed each containing 3 consonants and 2 vowels?

Answer 1

GANESHPURI: No. of letters $=10$ vevels are AEUI i.e. 4 and consonants 6.
Each word is containing 3 consonants out of 6 and 2 vowels out of 4.
We can select them in ${}^6{C}_3$ and ${}^4{C}_2$ ways. Thus the total number of combination (gr

oups) of 5 letters will be ${}^6{C}_3{\times }^4{C}_2$ by fundamental theorem.
But ${}^6{C}_3{\times }^4{C}_2=\frac{6!}{3!.3!}\times \frac{4!}{2!.2!}$
$=\frac{6\times 5\times 4}{\mathrm{3.2.1}}=\frac{4.3}{1.2}=120$ ways
Thus we have 120 groups each containing 5 leters i.e. 3 consonants and 2 vowels. Now the 5 letters in each group can be arranged amongst themselves in $5!4$ ways. i.e. 120 ways. Hence the total number of different words will be $120\times 120=14400$, by fundamental theorem.