Question

Anagrams are made by using the letters of the word ${}^{\prime }HINDUSTA{N}^{\prime }$ In how many of these anagrams none of the vowels come together?

Answer 1

$HINDUSTAN$

Number of letters $=9$

$N$ is repeated twice

Number of vowels $=3$

Number of consonants $=6$

Arranging $6$ Consonants

Number of ways $=\frac{6!}{2!}$ Ways

There are $7$ places for vowels

and there are $3$ vowels

Number of ways ${=}^7{P}_3$

Total ways of arranging ${=}^7{P}_3\times \frac{6!}{2!}$

$=210\times \frac{6!}{2!}$

$=75600$