Find the number of different 8-letter arrangement that can be made from the letters of the word DAUGHTER so that(i)all the vowels occur together(ii)all the vowels do not occur together

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Solution

(1) all vowels occur together.
Total number of letter in DAUGHTER = 8
vowels in Daughter = A,U&,E
since all vowels occur together
Assume AUE as single object
So are word becomes AUE ,D, 4, H, T, 3
Now arranging 3 vowels
= $^{3} P_{3}$
$= \frac{3!}{(3 - 3)!}$
= 3!
= 6 way
arranging 6 letters
Numbers we need to arrange
5 + 1 = 6
= $^{6} P_{6}$
= 6 !
= 720
Total No of average meets
$= 720 \times 6$
$= 7320$
(1) all vowel do not occur together
total number of permutations - number of permutations all occur come together
total permutation $=^{8} P_{8}$ = 40320
40320 - 4320 = 36000

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Similar questions

Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that

1) all vowels occur together

2) all vowels do not occur together.

8 -

D A U G H T E R

(i)

(i i)

Find the no.of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that:

(a)All vowels occur together.

(b)All vowels do not occur together.

(A)

(B)

8

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