Find the probability of eight letter word that can be formed from BLASTING so that vowel come together?
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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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ANSWER
(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
=
(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
=720×6
=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
Answer:
number of alphabets in the word BLASTING = 8
If vowels are to come together, they should be considered as a system. Thus, now the number of words is effectively B,L,S,T,N,G,(AI).
Thus, number of ways for arranging these 7 alphabets = 7! = 1×2×3×4×5×6×7 = 5040
While considering (AI) as a system, their positions can still be changed amongst themselves. Thus, number of ways for arranging the two vowels = 2! = 1×2 = 2.
Thus total probability of arranging the 8 alphabet word B,L,A,S,T,I,N,G with two vowels (A,I) = 7! × 2! = 5040 × 2 = 10080.