Results 1 to 6 of 6

Thread: Combinatorics: Upper and lower bounds

  1. #1
    Newbie
    Joined
    Oct 2017
    From
    Canada
    Posts
    4

    Combinatorics: Upper and lower bounds

    In a class of students 21 play basketball, 32 students pay hockey, and 32 students play soccer. In addition we know that 12 students play both basketball and hockey, 14 students pay both basketball and soccer, 11 both hockey and soccer. Each student in this class plays at least one of the three sports. What is the smallest possible upper bound and the largest possible lower bound that can be determined with the given information for the number of students in this class?

    Any hints on how to approach this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,454
    Thanks
    2728
    Awards
    1

    Re: Combinatorics: Upper and lower bounds

    Quote Originally Posted by usersmuser View Post
    In a class of students 21 play basketball, 32 students pay hockey, and 32 students play soccer. In addition we know that 12 students play both basketball and hockey, 14 students pay both basketball and soccer, 11 both hockey and soccer. Each student in this class plays at least one of the three sports. What is the smallest possible upper bound and the largest possible lower bound that can be determined with the given information for the number of students in this class?
    Any hints on how to approach this?
    See your other post. Learn to draw Venn diagrams.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2017
    From
    Canada
    Posts
    4

    Re: Combinatorics: Upper and lower bounds

    Quote Originally Posted by Plato View Post
    See your other post. Learn to draw Venn diagrams.
    I can't see how that works in this case where we have incomplete information?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2010
    Posts
    2,967
    Thanks
    1140

    Re: Combinatorics: Upper and lower bounds

    Quote Originally Posted by usersmuser View Post
    I can't see how that works in this case where we have incomplete information?
    The lower or upper bound occurs when a maximum number of students play all 3 sports. You cannot have more than 11 students play all 3 sports. That's because only 11 students play both hockey and soccer. Then you have the opposite bound when zero students play all three sports. As Plato suggested, do the Venn diagrams.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2010
    Posts
    2,967
    Thanks
    1140

    Re: Combinatorics: Upper and lower bounds

    Quote Originally Posted by SlipEternal View Post
    The lower or upper bound occurs when a maximum number of students play all 3 sports. You cannot have more than 11 students play all 3 sports. That's because only 11 students play both hockey and soccer. Then you have the opposite bound when zero students play all three sports. As Plato suggested, do the Venn diagrams.
    Oops, you need a minimum of 5 students that play all 3 sports because you have 21 basketball players, 12 that play basketball and hockey and 14 that play basketball and soccer. So, you need at least 5 players that play all three sports.
    Last edited by SlipEternal; Oct 19th 2017 at 08:08 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    742
    Thanks
    341

    Re: Combinatorics: Upper and lower bounds

    Combinatorics: Upper and lower bounds-bhs.jpeg
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Upper and lower bounds
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Sep 16th 2014, 01:45 PM
  2. Upper and Lower Bounds
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Sep 4th 2010, 01:04 PM
  3. Upper and lower bounds help?
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Feb 5th 2010, 02:07 PM
  4. Upper and Lower Bounds
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Aug 31st 2009, 06:52 PM
  5. Greatest lower bounds and least upper bounds
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Mar 31st 2008, 07:31 PM

/mathhelpforum @mathhelpforum