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Math Help - Permutation of word "UNBIASED" with condition

  1. #1
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    Permutation of word "UNBIASED" with condition

    Consider the word UNBIASED. How many words can be formed with the letters of the
    word in which no two vowels are together?

    If we arrange consonant first it will be:-

    4*3*2*1 = 24

    Now we are arrange 4 vowels as:-

    X C X C X C X C X

    Permutation of 5 objects taken 4 at a time. 5! = 120

    Answer is coming to be 120 * 24 = 2880

    But the actual answer is different.
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  2. #2
    Senior Member BAdhi's Avatar
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    What have you mentioned as X here?

    this is my attempt check whether this is correct

    If V is a vowel, and N is a non-vowel, the arrangment should be,

    V,N,V,N,V,N,V,N or N,V,N,V,N,V,N,V

    so for vowels no of arrangments - 4!
    for non-vowels no of arrangments - 4!

    total no. of arrangments = (4!.4!)2
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  3. #3
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    Not very sure but let me try.

    The total number of permutations of the letters in the word "UNBIASED" are 8!

    1. Now consider first event: take all the vowels together and make it one word
    (UIAE) N B S D. These 5 objects can be arranged in 5! ways. But again the 4
    vowels can be arranged in 4! ways so total
    5! * 4! = 120 * 24 = 2880

    2. The event where no two vowels are together, is a mutual exclusive event to the
    1st one. So total number of arrangements where no two vowels are together
    8! - (5!*4!) = 40320 - 2880 =37440

    Is that the answer you are looking for?
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  4. #4
    Senior Member BAdhi's Avatar
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    I think you've only excluded the arrangments where all vowels are together, so what about arrangments like UIBNASED and UIEBNASD
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  5. #5
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    Quote Originally Posted by vivek_master146 View Post
    Consider the word UNBIASED. How many words can be formed with the letters of the
    word in which no two vowels are together?
    Answer is coming to be 120 * 24 = 2880
    But the actual answer is different.
    The four consonants create five places to put the four vowels.
    \underline{\quad}C\underline{\quad}C\underline{\qu  ad}C\underline{\quad}C\underline{\quad}
    We can have two consonants together.
    So we get 5\cdot(4!)^2=2880

    Why do you say the actual answer is different from that?
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  6. #6
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    Thanks plato. I also computed it in the same say. The answer given in the book is 1152. So i was confused! R u sure that this is the correct answer. ?
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  7. #7
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    Quote Originally Posted by vivek_master146 View Post
    The answer given in the book is 1152. So i was confused! R u sure that this is the correct answer. ?
    The book's answer is the same as reply #2.
    BUT, that assumes that no two consonants are adjacent.
    That is not the way the question reads.
    I suspect that whoever worked out that answer assumed that even though it is not correct. Because it does not count UNBASIDE for example.
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