sir / mam pls tell the way to solve---- From the word HINDUSTAN find out---- (a) How many anagrams can - Maths - Permutations and Combinations

Question

sir / mam pls tell the way to solve----
From the word HINDUSTAN find out----
(a) How many anagrams can be made by
using the letters of the word HINDUSTAN?  
(b) How many of these anagrams begin and
end with a vowel
(c) In how many of these anagrams all the
vowels come together
(d) In how many of these anagrams none of
the vowels come together
(e) In how many of these anagrams do the
vowels and the consonants occupy the same relative positions as in
HINDUSTAN?

Ankush Jain · Accepted Answer

In the word 'HINDUSTAN', the letter N repeats itself two times a) Total number of anagrams formed = Arrangements of nine letters taken all at a time $= \frac{9!}{2!} = 181440$ (b) In the word 'HINDUSTAN' there are 3 vowels and 6 consonants, in which 2 consonants(N) are alike Now, 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 $= 3 \times 2 \times \frac{7!}{2!} = 15120$ (c) Assume the vowels (I, U, A) as a single letter The letters (IUA) H, D, S, T, N, N can be arranged in $\frac{7!}{2!}$ ways Also IUA can be arranged among themselves in 3! = 6 ways Hence the total number of anagrams $= \frac{7!}{2!} \times 6 = 15120$ Due to paucity of time it would not be possible for us to provide the answers to all your queries We are providing solutions to some of your queries Try solving rest of the questions yourself and if you face any difficulty then do get back to us