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.
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.
975 helpful votes in Math .
Total number of words formed from the word DAUGHTER = 8! = 40320 words
.
(1) There are 3 vowels in DAUGHTER - A,U,E.... If vowels are tied together and used as single letter then:-
(AUE)DGHTR
.
Here the brackets represent them to be together. Hence now the no. of words formed will be = 6! * 3!, since AUE can be arranged themselves in 3!.
.
So if all vowels occur together then the no. of words formed = 6! * 3! = 720 * 6 = 4320 words .
.
(2) The no. of words formed if vowels do not occur together = Total no. of words formed from DAUGHTER - No. of words formed when vowels occur together.
Henceno. of words formed if vowels do not occur together = 8! - (6!*3!) = 40320 - 4320 = 36000 words.
.
Hence answers:-
(1) 4320 words
(2) 36000 words
.
.
Total number of words formed from the word DAUGHTER = 8! = 40320 words
.
(1) There are 3 vowels in DAUGHTER - A,U,E.... If vowels are tied together and used as single letter then:-
(AUE)DGHTR
.
Here the brackets represent them to be together. Hence now the no. of words formed will be = 6! * 3!, since AUE can be arranged themselves in 3!.
.
So if all vowels occur together then the no. of words formed = 6! * 3! = 720 * 6 = 4320 words .
.
(2) The no. of words formed if vowels do not occur together = Total no. of words formed from DAUGHTER - No. of words formed when vowels occur together.
Henceno. of words formed if vowels do not occur together = 8! - (6!*3!) = 40320 - 4320 = 36000 words.
.
Hence answers:-
(1) 4320 words
(2) 36000 words
.
