Question

How many anagrams can be made by using the letters of the word HINDUSTAN?
How many of these anagrams begin and end with a vowel?

Answer 1

Total number of anagrams $=$ Arrangements of nine letters taken all at a time

$=\frac{9!}{2!}$

$=181440$

We have 3 vowels and 6 consonents, in which 2 consonents are alike

The first place can be filled in 3 ways and the last in 2 ways

The rest of the places can be filled in $\frac{7!}{2!}$ ways.

Hence the total number of anagrams $=3\times 2\times \frac{7!}{2!}$

$=15120$