Question

The word DIFFERENTIATE is arranged in a circle. In how many ways can it be arranged such that the vowels are in the same order as they are in the original word when seen clockwise, and such that no 2 vowels appear together?

Answer 1

@DLS @shubhamsrg

Answer 2

OMG. This is an interesting question.

Answer 3

How about we remove the vowels first?
DIFFERENTIATE
DFFRNTT

Answer 4

And how many ways can that be arranged in a circle?

Answer 5

6!/2?

Answer 6

6! *

Answer 7

That was an honest question @RnR I don't know... researching... please wait
^.^

Answer 8

it is 6!, i just rechecked.

Answer 9

Taking into account that there are two F's and two T's?

Answer 10

Okay, the formula for a circular permutation of n elements is
(n-1)!
But if there are repeated digits, you divide it by
\[\Large \frac{(n-1)!}{a_1!a_2!...}\]If there are a_1 repeated entries of the same entry, and a_2 repeated entries of another entry...etc

Answer 11

Yes, i made a mistake there.

Answer 12

So, that means you divide 6! by 4, right? or 2!2!

Answer 13

Should be easy enough, 720 divided by 4 is 180

Answer 14

Now, the hard part... the vowels...

Answer 15

It's a good thing the vowels are forbidden from being next to each other, that simplifies things :D

Answer 16

I'm sorry. There is some problem with my internet connection.

Answer 17

So now the task is how many ways you can "sprinkle" the vowels into the permuted consonants...

Answer 18

There are seven consonants arranged in a circle, means there are seven "spaces" between them, right?

Answer 19

I guess.

Answer 20

Let's call the spaces 1,2,3,4,5,6, and 7
And you have five vowels to distribute among those spaces, so let's count how many ways to do that...
First is to have the first vowel on space 3, and the rest following Remember that the order of the vowels has to be preserved, so this is just one way.

Answer 21

Wait, scratch that, I keep forgetting that this is in a circle... sorry

Answer 22

I think there will be 6 places left for vowels with two consonets coming together.

Answer 23

consonants*

Answer 24

There will be seven spaces because there are seven consonants... ARRANGED IN A CIRCLE

Answer 25

Right

Answer 26

Catch me so far?
NOTE: There may be an actual formula for this, but if there is, I don't know it, I'm trying to deduce it using just intuition :)

Answer 27

So, you have five vowels, and since this is arranged in a circle, the first vowel may be put in any of the seven spaces, right?

Answer 28

6 vowels? d I ff E r E nt I A t E

Answer 29

Right. 6 sorry.. Good thing, too. Makes it easier...

Answer 30

So, we place the first vowel.... I, in one of the seven spaces, let's call THAT space 1, ok?

Answer 31

So 7c6 would be the answer for vowels no?

Answer 32

divided by the repeating vowels. Ah, i keep forgetting that.

Answer 33

I don't know... let's check? I don't use formulas blindly if I can avoid it :D

Answer 34

Ok. It would be better.

Answer 35

First, the case where the first vowel, I, is in space 1.
Then the thing to count...
how many ways can you put the remaining five vowels into the six remaining spaces?

Answer 36

Maintaining order, mind you.

Answer 37

Oh, i completely missed the order thing. Sorry.
So there are 3 E's , 3c1?

Answer 38

Well, you have six remaining spaces
_ _ _ _ _ _ and five vowels E E I A E
You must maintain that arrangement while distributing these among the six spaces...
For example
EE_IAE (the space indicates a space was skipped, in this case, no vowel in space 4.

Answer 39

Now, take notice, when choosing which spaces of the six you place your five letters, you are, in effect, choosing ONE SPACE where NOT to put a letter. ;)

Answer 40

IE if you want to be thorough, the arrangements are
_EEIAE
E_EIAE
EE_IAE
EEI_AE
EEIA_E
EEIAE_

Answer 41

Got it so far?

Answer 42

Yup. I think i did.

Answer 43

So... with the first vowel, I, in space number 1, we have six different ways to 'sprinkle' the vowels among the non-vowels...

Answer 44

BUT that's only if the first vowel I is in space number 1. in space number 2, for instance... or 3...etc... what then? ;)

Answer 45

wait, can we even have I at 1? Shouldnt an E be there? The order?

Answer 46

No. I is your first vowel. We started with E because "I" was already placed :D

Answer 47

d I ff E r E nt I A t E

Answer 48

Oh, yeah. Sorry.

Answer 49

Okay. So, if I is placed at space number 1, we have six combinations.
What if it's placed in another space, besides space number 1?

Your thoughts, @RnR ? :)

Answer 50

Hmm i dont think i know.

Answer 51

Okay... let's exploit the power of VISUAL AIDS!!!
>:D

but seriously though... if the initial I is placed here...

Then there are 6 different ways to arrange the rest of the vowels here...