Results 1 to 4 of 4
Like Tree2Thanks
  • 1 Post By DenisB
  • 1 Post By Plato

Thread: Permutations of the word ARRANGEMENT. A level question.

  1. #1
    Member
    Joined
    May 2009
    Posts
    141
    Thanks
    1

    Permutations of the word ARRANGEMENT. A level question.

    Hi folks,

    Here is a question from my A level maths text book. I can't get the right answer!
    find the total number of 4 letter permutations of 4 letters selected from the word ARRANGEMENT. How many of these permutations contain 2 Rs?

    Here are my calculations, can anyone spot the error?

    We have 11 letters with 4 pairs (2A, 2E, 2N, 2R) and three singles G, M, T. Consider the mutually exclusive scenarios and add them up.

    1) Arrangements of 4 letters in which all are different. i.e. select 4 letters from the 7 different ones.
    here goes with the LaTex! hope it works.
    [LaTeX ERROR: Convert failed] = 7!/(7 - 4)! = 7!/3! = 7 x 6 x 5 x 4 = 840 permutations.

    2) Arrangements of 4 letters with one pair and two other letters
    We select a pair, say the 2 Rs in one way. Then we select two letters from the available 6
    i.e. [LaTeX ERROR: Convert failed] = 6! / 4! = 6 x 5 = 30
    These 4 letters can be arranged in 4! / 2! ways = 4 x 3 = 12 ways
    so we have 30 x 12 = 360 arrangements.
    There are 4 pairs so the total number of 4 letter arrangements gives 360 x 4 = 1440 permutations

    3) Arrangements of 4 letters containing 2 pairs
    In this case we select 2 letters (each a pair) from the available 4 pairs i.e. [LaTeX ERROR: Convert failed]
    = 4! / 2! = 4 x 3 = 12 ways. These 4 letters can be arranged in 6 ways for example
    AAEE, EEAA, AEAE, EAEA, AEEA, EAAE
    so we have 12 x 6 = 72 permutations

    So the total number of permutations should be 840 + 1440 + 72 = 2352

    The number of permutations containing 2 Rs is taken directly from part 2) i.e. 360.

    Both parts are wrong. The correct answers are 1596 and 198.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    1,634
    Thanks
    304

    Re: Permutations of the word ARRANGEMENT. A level question.

    May be easier if digits used: arrangement = 12213456537
    (or 11223344567)

    Go here to see a similar problem:
    https://www.quora.com/Find-the-arran...rd-MATHEMATICS
    Thanks from s_ingram
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,149
    Thanks
    2603
    Awards
    1

    Re: Permutations of the word ARRANGEMENT. A level question.

    Quote Originally Posted by s_ingram View Post
    Here is a question from my A level maths text book. I can't get the right answer!
    find the total number of 4 letter permutations of 4 letters selected from the word ARRANGEMENT.
    How many of these permutations contain 2 Rs?
    There are four pairs {2A, 2E, 2N, 2R} and three singles G, M, T.
    You must list the possible contents:
    1. All four different letters from seven: $^7\mathcal{P}_4$
    2. two letters the same & two different: ${4\choose 1}{6\choose 2}\dfrac{4!}{2!}$
    3. two pair of the same letter: ${4\choose 2}\dfrac{4!}{(2!)^2}$


    Can you explain?
    Thanks from s_ingram
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    May 2009
    Posts
    141
    Thanks
    1

    Re: Permutations of the word ARRANGEMENT. A level question.

    Many thanks to Plato and DenisB.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 9
    Last Post: Mar 1st 2014, 01:08 PM
  2. Permutations word problem check
    Posted in the Statistics Forum
    Replies: 1
    Last Post: Mar 30th 2011, 05:04 PM
  3. AS Level Maths question.
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Oct 15th 2010, 09:50 AM
  4. Jordan Curve (high level, graduation level)
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Apr 29th 2009, 05:24 AM
  5. rate/water level word problem
    Posted in the Calculus Forum
    Replies: 9
    Last Post: Mar 29th 2008, 07:07 PM

/mathhelpforum @mathhelpforum