Math Questions and Answers

Start Your Free Trial

In how many ways can the letters of the word accommodation be arranged if all the letters are used and all the vowels are together?

Expert Answers info

Tushar Chandra eNotes educator | Certified Educator

calendarEducator since 2010

write12,554 answers

starTop subjects are Math, Science, and Business

There are 13 letters in the word ACCOMMODATION. The number of unique letters is 8, with 2 A, 2 C, 3 O and 2 M.

There are 6 vowels and 7 consonants. If the number of vowels is kept together, keeping in mind the...

(The entire section contains 102 words.)

Unlock This Answer Now

check Approved by eNotes Editorial

llltkl | Student

ACCOMMODATION is a thirteen letter word.

There are 6 vowels in it. Taking all the vowels together as a single unit, there would be 2 Cs, 2 Ms, 1D,  1 T and 1 N, i.e. 8 units altogether, including several repetitions.

There can be `(8!)/(2!*2!)` different arrangements possible with these words.

Again, among the 6 vowels there are 3 Os and 2 As. They can be arranged in `(6!)/(3!2!)` different ways.

Therefore, the number of ways the letters of the word “accommodation” can be arranged, if all the letters are used and all the vowels are together, is

`((8!)/(2!*2!))* ((6!)/(3!2!))= (8!*6!)/(3!*2!*2!*2!)=604800` 

aruv | Student

A-2 ,C-2, D-1,I-1,M-2,N-1, O-3 and T-1

Three vovels A ,I and O

Let consider all  vovels  as X=AIO

Now we have 6 letters  , X , C , D , M , N , T  and these letters can be arranged and form  `(7!)/(2!xx2!)=1260` words

Three vowels together can be arranged and form   `(6!)/(2!xx3!)`

`=60` words

By Fundamental principle of counting all letters are usedand all vowels are together form  `60xx1260=75600`  words