In how many ways we can arrange the word TEACHER such that no two vowels should be together.

Question

In how many ways we can arrange the word TEACHER such that no two vowels should be together.

Answer 1

Hi Krishna,

 

We need to break the word down into V - Vowels and C - Consonants

 

And then look at the patterns that can be created with V and C without having two vowels next to each other

 

VCVCVCC

CVCVCVC

CCVCVCV

VCCVCVC

VCCVCCV

VCVCCVC

VCVCCCV

VCCCVCV

 

I think this respresents all possible arrangements of the vowels and consonants without two vowels together

 

Now for each arrangement we can arrange the Consonants 4*3*2 ways  or 24 different ways

and for each arranagement we can arrange the Vowles 3*1 ways (not 3*2) since the E is repeated and would not show a difference when moved to the same spot as the other E

 

therefore, for each arrangement we had 24*3 = 72 possible ways of arranging the letters

 

and we have 8 different arrangements that work

 

so our total would be

 

8*72 = 576 possible different ways to arrange the word without having vowels next to each other

 

Hope this helps

Answer 2

I get 180 different ways.