Question

In how many ways the letters of the word ‘TEACHERS’ be arrange so that

(i) All vowels come together.

(ii) All vowels come together and all consonants come together.

(iii) There is always a gap of 3 letters between both E’S.

(iv) The word starts with T and ends with S?​

Answer 1

Given:

Letters of the word ‘TEACHERS’ .

To Find:

(i) All vowels come together.

(ii) All vowels come together and all consonants come together.

(iii) There is always a gap of 3 letters between both E’S.

(iv) The word starts with T and ends with S?​

Solutions:

Here, in 'teacher' -

(i) :- There are three vowels A , E and E.

So, taking all the vowels as one letter we have, 5 letter word to arrange.

So we can arrange 5 letter word in 5! ways , but we already have two E so, we have to divide it by 2!

Hence,

letter of ‘TEACHERS’ be arrange so that all vowels come together

            = 5!/2!

            =60 ways.

(ii) :-  There are three vowels A , E and E and 5 consonants T,C,H,R and S.

So, taking all the vowels as one letter and all consonants as one word  we have, 2 letter word to arrange.

So, we can arrange 2 letter word in 2! ways, but we already have two E so, we have to divide it by 2!

Therefore, letter of ‘TEACHERS’ be arrange so that all vowels  and all consonants come together = 2!/2! = 1 ways.

(iii) :- There is always a gap of 3 letters between both E’s. = 4!= 24

(iv) :- We have to arrange the letter of the word 'teacher' starts with T and ends with S.

So, we have to fix T and S at 1st and 8th position respectively so, we have 6 letter word to arrange.

So, we can arrange 6 letter word in 6! ways, but we already have two E so, we have to divide it by 2!

Therefore, no. of ways to arrange the word teacher starts with T and ends with S = 6!/2! =360