Results 1 to 2 of 2

Math Help - generating functions again! :(

  1. #1
    Junior Member
    Joined
    Nov 2009
    Posts
    37

    Exclamation generating functions again! :(

    Heather just had new neighbours move in next door and would like to make a fruit basket for
    them. When her husband returned from the grocery store he had a bag containing apples,
    bananas, and oranges. How many different possibilities are there for a fruit basket consisting
    of exactly 36 pieces of fruit if
    (a) there is an ample supply of apples and bananas, but there are only 4 oranges?
    (b) there are 16 of each kind of fruit?
    (c) there is an ample supply of bananas and oranges, and there are 4 distinguishable apples
    (one Fuji, one Macintosh, one Spartan, and one Ambrosia)?

    (In each case, fruit baskets are distinguished
    solely by how many of each type of fruit they contain.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by sbankica View Post
    Heather just had new neighbours move in next door and would like to make a fruit basket for
    them. When her husband returned from the grocery store he had a bag containing apples,
    bananas, and oranges. How many different possibilities are there for a fruit basket consisting
    of exactly 36 pieces of fruit if
    (a) there is an ample supply of apples and bananas, but there are only 4 oranges?
    (b) there are 16 of each kind of fruit?
    (c) there is an ample supply of bananas and oranges, and there are 4 distinguishable apples
    (one Fuji, one Macintosh, one Spartan, and one Ambrosia)?

    (In each case, fruit baskets are distinguished
    solely by how many of each type of fruit they contain.)
    I'll get you started on (a).

    Let a_r be the number of ways to prepare a fruit basket with r pieces of fruit. We want to find the Ordinary Power Series Generating Function (OPSGF) f(x) = \sum_r a_r x^r.

    The OPSGF for the number of apples is
    1 + x + x^2 + \dots = (1-x)^{-1},

    the series for the number of bananas is the same,
    and the series for the number of oranges is
    1 + x^2 + x^3 + x^4 = 1 + x^2 + x^3 + \dots - (x^5 + x^6 + x^7 + \dots) = (1-x)^{-1} - x^5 \; (1-x)^{-1}

    So
    f(x) = (1-x)^{-1} \cdot (1-x)^{-1} \cdot [ (1-x)^{-1} - x^5 \; (1-x)^{-1} ]
    = (1 - x^5) \; (1-x)^{-3}

    The number of ways to get 36 pieces of fruit in the basket is the coefficient of x^{36} when f is expanded as a series.

    Do you see how to find this coefficient?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Generating functions
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: December 8th 2011, 04:30 AM
  2. generating functions
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 1st 2010, 01:15 AM
  3. generating functions
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: October 24th 2009, 07:01 AM
  4. Generating Functions
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: April 25th 2008, 04:43 PM
  5. Generating functions...need some help here
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 31st 2008, 04:32 PM

Search Tags


/mathhelpforum @mathhelpforum