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From the word HINDUSTAN =====
(a) In how many of these anagrams none of the vowels come together.
(b) In how many of these anagrams do the vowels and the consonants occupy the same relative positions as in HINDUSTAN?

  • maths-probability -

    number of arrangements without restriction = 9!/2 = 181440 (There were 2 N's, so we divide by 2!)

    We have 3 vowels, we don't want any 2 to be beside each other.
    so consider the possible pairs.
    IU IA UA UI AI AU , that is there are 6 of the, or P(3,2)

    So if we consider a pair of vowels to be X , the number of ways for the vowels to be together is
    8! (6)/2 = 120960
    (notice that would include the cases where the 3 vowels are together
    e.g. IAHNDUSTN

    number of ways for the vowels to be apart
    = 181440 - 120960 = 60480

    b) The relative position of vowels and consonants must remain the same, so we need

    CVCCVCCVC , given 6 cosonants (2 the same) and 3 vowels
    or 6x3x5x4x2x3x2x1x1/2! = 2160

    check my arithmetic

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