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Respected Sir, I have one problem in permutation and combination. Hope you will help. My question is : " In how many ways we can arrange the letters of the word ‘ TEACHER ’ such that no two vowels are together. " Please give full explanation. Thanks & Regards, Krishna

Respected Sir,
 
I have one problem in permutation and combination.
 
Hope you will help. 
 
My question is : 
 
 " In how many ways we can arrange the letters of the word  ‘ TEACHER ’ such that no two vowels are together." 
 
Please give full explanation.
 
Thanks & Regards,
Krishna

Grade:12th pass

2 Answers

Hemanth Sai
44 Points
9 years ago
There are total 7 lettersout of which 3 are vowels among which 2 are repeated
first the total numbers of cases are 7!/2 (bcz 2 ltrs rptd)
now let us caculate the number with vowels to gather = 5!*3!/2!
 
Now subtract the first one from second and thats the ans 
Yash Jain
55 Points
9 years ago
For the given word ‘TEACHER’, we have 3 vowels(V), in which 2 are distinct, and 4 consonants(C).
Now we have to arrange them in such a way that vowels are never together. On interpreting the condition given, the required format of words is as follows:
_C_C_C_C_  any 3 out 5 ‘_’ can be filled by vowels.
This arrangement can be achieved by the following steps:
1. All the consonants can be arranged in 4! ways.
2. All the ‘_’ can be filled by vowels in 5P3/2! ways(because of 2E’s).
Then, using the multiplication principle, we have our answer as
No. of ways = 24x5x4x3/2 = 720
I hope you got it now.
PS: Please inform me if I am wrong somewhere.

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