From the word HINDUSTAN =====

(a) In how many of these anagrams none of the vowels come together.
(b) In how many of these anagrams do the vowels and the consonants occupy the same relative positions as in HINDUSTAN?

number of arrangements without restriction = 9!/2 = 181440 (There were 2 N's, so we divide by 2!)

We have 3 vowels, we don't want any 2 to be beside each other.
so consider the possible pairs.
IU IA UA UI AI AU , that is there are 6 of the, or P(3,2)

So if we consider a pair of vowels to be X , the number of ways for the vowels to be together is
8! (6)/2 = 120960
(notice that would include the cases where the 3 vowels are together
e.g. IAHNDUSTN
and IAUHNDSTN

number of ways for the vowels to be apart
= 181440 - 120960 = 60480

b) The relative position of vowels and consonants must remain the same, so we need

CVCCVCCVC , given 6 cosonants (2 the same) and 3 vowels
or 6x3x5x4x2x3x2x1x1/2! = 2160

check my arithmetic

To answer your questions about anagrams of the word HINDUSTAN:

(a) In how many of these anagrams none of the vowels come together:
To solve this, we need to find the total number of anagrams formed from the letters of the word HINDUSTAN and then subtract the cases where the vowels come together.

The word HINDUSTAN has 9 letters - H, I, N, D, U, S, T, A, N.
It has 4 vowels - I, U, A, and the remaining 5 letters are consonants - H, N, D, S, T.

Now, let's find the total number of anagrams of HINDUSTAN:
The total number of ways to arrange the 9 letters is 9! (factorial), which is equal to 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880.

Next, let's find the number of cases where the vowels come together (none of the vowels come together):
If we treat the 3 vowels (I, U, A) as a single unit, we have 7 units to arrange (H, N, D, S, T, Vowel unit, N). The vowel unit can be arranged in 3! = 6 ways.

However, within the vowel unit, the vowels themselves can be rearranged in 3! = 6 ways.

So, the total number of cases where the vowels come together is 6 x 6 = 36.

Finally, the number of anagrams where none of the vowels come together is obtained by subtracting the cases where the vowels come together from the total number of anagrams:
Total anagrams without restrictions - Cases where vowels come together = 362,880 - 36 = 362,844.

Therefore, there are 362,844 anagrams of HINDUSTAN where none of the vowels come together.

(b) In how many of these anagrams do the vowels and the consonants occupy the same relative positions as in HINDUSTAN:
To solve this, we need to find the total number of anagrams where the vowels and consonants occupy the same relative positions.

There are 4 vowels and 5 consonants in HINDUSTAN.
To keep the same relative positions, we need to arrange the vowels (I, U, A) among themselves and the consonants (H, N, D, S, T) among themselves.

The number of ways to arrange the vowels among themselves is 3! = 6.
The number of ways to arrange the consonants among themselves is 5! = 120.

Thus, the total number of anagrams with the vowels and consonants in the same relative positions is 6 x 120 = 720.

Therefore, there are 720 anagrams of HINDUSTAN where the vowels and the consonants occupy the same relative positions.

To answer these questions, we need to break them down step by step.

(a) In how many of these anagrams none of the vowels come together?

To solve this question, we first need to determine the total number of anagrams that can be formed from the word "HINDUSTAN."

The word "HINDUSTAN" has 9 letters in total. However, it contains repeated letters: there are 2 "N's" and 2 "I's."

The formula to calculate the number of anagrams when there are repeated letters is:

Total number of anagrams = (Total number of letters)! / (Number of times each letter is repeated1!*Number of times each letter is repeated2!...)

In this case, we have 9 letters with 2 "N's" and 2 "I's":

Total number of anagrams = 9! / (2! * 2!) = 9! / 4 = 9 * 8 * 7 * 6 * 5 = 15,120

Now, we need to determine the number of anagrams where none of the vowels come together. For this, we can treat the vowels (I, U, A, E) as individual entities.

We have 4 vowels, so we can consider them as separate objects. Thus, the number of ways to arrange them without any constraints is:

Number of ways to arrange the vowels = 4! = 4 * 3 * 2 * 1 = 24

However, since we don't want any vowels to be together, we need to subtract the cases where there are consecutive vowels:

Number of ways to arrange the vowels with consecutive vowels = 3! = 3 * 2 * 1 = 6

Now, to find the number of anagrams where none of the vowels come together, we subtract the arrangements with consecutive vowels from the total number of arrangements:

Number of anagrams with no consecutive vowels = Total number of anagrams - Number of arrangements with consecutive vowels
= 15,120 - 6 = 15,114

Therefore, there are 15,114 anagrams of "HINDUSTAN" where none of the vowels come together.

(b) In how many of these anagrams do the vowels and the consonants occupy the same relative positions as in HINDUSTAN?

To solve this question, we need to understand that we are looking for anagram arrangements where the vowels (I, U, A, E) and the consonants (H, N, D, S, T) have the same relative positions as in the original word "HINDUSTAN."

To calculate this, we need to consider the number of anagrams we calculated previously, where none of the vowels come together (15,114).

Now, as the vowels and consonants need to occupy the same relative positions, we can consider the vowels and consonants as two separate groups. Therefore, the number of anagrams where the vowels and consonants have the same relative positions is:

Number of anagrams with vowels and consonants in the same relative positions = (Number of ways to arrange the vowels) * (Number of ways to arrange the consonants)

The number of ways to arrange the vowels and consonants separately can be calculated as:

Number of ways to arrange the vowels = 4!
Number of ways to arrange the consonants = 5!

Using the factorial formula, we can calculate:

Number of ways to arrange the vowels = 4! = 4 * 3 * 2 * 1 = 24
Number of ways to arrange the consonants = 5! = 5 * 4 * 3 * 2 * 1 = 120

Now, we multiply these two together to get the desired result:

Number of anagrams with vowels and consonants in the same relative positions = Number of ways to arrange the vowels * Number of ways to arrange the consonants
= 24 * 120 = 2,880

Therefore, there are 2,880 anagrams of "HINDUSTAN" where the vowels and the consonants occupy the same relative positions.