Question

In how many ways can the letters of the word permutation be arranged if all vowels are together and there are always 4 letters between p and s

Answer 1

The word here is

PERMUTATIONS

it is composed of

P ….(1)

E ….(1)

R ….(1)

M ….(1)

U ….(1)

T …(2)

A …(1)

I ….(1)

O …(1)

N …(1)

S ….(1)

So all alphabets except T is unary. Total there are 12 alphabets.

P and S contains 4 aphabets within. So if we exclude P and S from 12, there will be 12–2=10 aphabets remaining , that can sit between P and S.

From these 10 we need to select 4 alphabets which can be done in 10c4 = 210 ways.

Let us assume the 4 alphabets guarded by P and S be denoted as another new alphabet ¥>

Now we divide this selection in 3 cases depending upon the composure of ¥:

Case 1 ( contains two T's) :

Here the 4 numbers have P and S in both end.

So P and S and these 4 alphabets can be considered as a unit ..say ¥.

So ¥ and remaining 12-(4+2)=6 alphabets

ie. total 6+1=7 alphabets

can arrange within themselves in 7! ways.

For each of these 7! permutations,

the 4 aphabets guarded by P & S can permute within themselves in 4!/(2!) = 12 ways. We are dividing by 2 as there are two T's. Also P and S may interchange positions.

So total cases of arrangement = 7! * 12*2

Case 2 ( contains one T's) :

Here everything remains as above. Only the permutations of 4 elements guarded by P and Q don't get divided by 2.

So total cases of arrangement = 7! * 24*2

Case 2 ( contains no T's) :

Here the two Ts are outside. So here ¥ and remaining 6 can arrange within themselves in 7!/2 ways as the set other than ¥ contains two Ts.

So total arrangements here = (7!/2)*4!*2= 7!*24

So total combining the 3 cases is

(7!*24) + (7!*48) + (7!*24)

= 7!*96

=483840

Answer 2

Answer:

2419200

Step-by-step explanation:

P E R M U T A T I O N S

Here, the vowels A,E,I,O,U must be taken as 1 group ( 1 letter )

( As if A=E=I=O=U)

Then T repeats 2 times, so 2t must be taken as 1 group ( 1 letter )

Therefore remains 6 letters + 1 ( 2t's) + 1 (A,E,I,O,U)

= 8

The 5 vowels again can be arranged in 5! ways

Therefore it can be arranged in

(8! / 2!) x 5! = 2419200

Hope it helps, if u didn't understand, just leave a comment, I'll explain it more deeply...