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Math Help - Problem Solving 2 - eyebrow

  1. #1
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    Problem Solving 2 - eyebrow

    Well i think ill just go ahead and make a separate thread for each question. Please could you post your worked out solutions and anything else which would help me understand how to solve the following problem. Thanks

    Q2. An eyebrow is an arrangement of the numbers 1,2,3,4 and 5 such that the second and fourth numbers are each bigger than both their immediate neighbours. For example, (1,3,2,5,4) is an eyebrow and (1,3,4,5,2) is not
    The number of eyebrows is

    a) 16 b)12 c)15 d) 24 e)18
    Answer: A
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  2. #2
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    Q2. An eyebrow is an arrangement of the numbers 1,2,3,4 and 5 such that the second and fourth numbers are each bigger than both their immediate neighbours. For example, (1,3,2,5,4) is an eyebrow and (1,3,4,5,2) is not

    the number 5 can be in the 2nd and 4th position since it's the largest number. when this occur.

    when 5 is in the 2nd position, 4 must be in 4th position.

    _ 5 _ 4 _

    that's 3! = 6 ways
    or vice versa = 6 x 2 =12 ways

    or the 4th position could be occupied by 3, providing that the number 4 is not next to 3.

    _ 5 _ 3 _

    4 has two places to go. and the remaining has 1. hence 2 x 1 = 2 ways
    and you can also have the alternate, where 3 sits in the 2nd position. hence 2 x 2 = 4 ways

    Total = 4 +12 = 16 ways
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    Quote Originally Posted by differentiate View Post
    Q2. An eyebrow is an arrangement of the numbers 1,2,3,4 and 5 such that the second and fourth numbers are each bigger than both their immediate neighbours. For example, (1,3,2,5,4) is an eyebrow and (1,3,4,5,2) is not

    the number 5 can be in the 2nd and 4th position since it's the largest number. when this occur.

    when 5 is in the 2nd position, 4 must be in 4th position.

    _ 5 _ 4 _

    that's 3! = 6 ways
    or vice versa = 6 x 2 =12 ways

    or the 4th position could be occupied by 3, providing that the number 4 is not next to 3.

    _ 5 _ 3 _

    4 has two places to go. and the remaining has 1. hence 2 x 1 = 2 ways
    and you can also have the alternate, where 3 sits in the 2nd position. hence 2 x 2 = 4 ways

    Total = 4 +12 = 16 ways
    Understood,
    thanks
    Last edited by 99.95; March 15th 2010 at 03:47 AM.
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