Question

How many six letter words containing 4 consonants and 2 vowels can be formed using the letters of the word GANESHPURI, so that the vowels occur together?

Answer 1

Hint: We need to break it in two parts of choosing and arranging. We first choose 4 consonants from 6 consonants and 2 vowels from 4 vowels and find the number of ways it can be done. Then we arrange them to find the number of arrangements possible. We multiply both to find the total number of possible outcomes.

Complete step by step answer:
The word GANESHPURI has 10 distinct letters. The distribution of vowels and consonants are 4 and 6 respectively.
Now we need to form a word of 4 consonants and 2 vowels following some conditions. This condition is that the vowels occur together.
So, first we complete the step to find the total 6 letters of 4 consonants and 2 vowels from 10 letters of GANESHPURI consisting 6 consonants and 4 vowels.
We are choosing 4 consonants from 6 consonants and 2 vowels from 4 vowels.
The consonant part can be done in ${}^{6}{{C}_{4}}$ ways and the vowel part can be done in ${}^{4}{{C}_{2}}$ ways.
Applying the theorem of combination, we get the number of ways will be ${}^{6}{{C}_{4}}\times {}^{4}{{C}_{2}}$.
Solving we get ${}^{6}{{C}_{4}}\times {}^{4}{{C}_{2}}=\dfrac{6!}{4!\times 2!}\times \dfrac{4!}{2!\times 2!}=90$.
Now we have got 6 distinct letters to form a word. We just need to be careful about two vowels sticking together.
So, we are arranging 6 terms out of which 2 are always together. We assume those two vowels as a unit.
So, there will be 5 total letters. We arrange them in $5!=120$ ways. But the unit of vowels can also arrange in between themselves in 2 ways by sticking together. So, the number of ways 6 distinct letters can be arranged to form a word is $120\times 2=240$.
Choosing the letters was done in 90 ways. For each of those ways the arrangement can be done in 240 ways. So, the total number of ways will be $90\times 240=21600$.

The number of six letter words is 21600.

Note: We need to be careful about not thinking about the vowels being together as the possible choices first need to be set straight. Vowels’ arrangement comes when we are dealing with the whole word of six letters.