Question

How many $6-$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

Number of ways to choose 4 constants out of 6${=}^6{C}_4$

Number of ways to choose 2 vowels out of 4${=}^4{C}_2$

As vowels are together, the word may be assumed to be a permutation of 5 letters leading to 5! ways

vowels may themselves permutate by 2! ways.

$Totalno.ofwordspossible{=}^6{C}_4{\times }^4{C}_2\times 5!\times 2!$
$=\frac{6!}{4!\times 2!}\times \frac{4!}{2!\times 2!}\times 5!\times 2!$
$=21600$