The Tennis Club Problem Supplementary Notes - PowerPoint PPT Presentation

The tennis club problem supplementary notes l.jpg
Download
1 / 7

  • 519 Views
  • Uploaded on
  • Presentation posted in: Sports / Games

The Tennis Club Problem Supplementary Notes. A tennis club has 2n members. We want to pair up the members (by twos) to play singles matches Question: In how many ways can we pair up all the members of the club?.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Presentation

The Tennis Club Problem Supplementary Notes

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


The tennis club problem supplementary notes l.jpg

The Tennis Club Problem Supplementary Notes

  • A tennis club has 2n members.

  • We want to pair up the members (by twos) to play singles matches

  • Question: In how many ways can we pair up all the members of the club?


Slide2 l.jpg

  • First, from the 2n members we have, choose 2 of them and put them on team 1

  • Then, from the remaining 2n-2 members, choose 2 of them and put them on team 2

  • Continue this way, at the ith step, choose 2 of the remaining 2n-2(i-1) members and put them on team i


Slide3 l.jpg

  • How many ways can we perform this process?

  • By the product principle, the number of ways is


Slide4 l.jpg

  • Is this the answer to the original question?

  • No

    • We have a double counting problem!

  • Let’s consider an example of the same problem with n = 2

    • The tennis club has 4 people and we want to pair them up by twos


Slide5 l.jpg

  • Suppose the members are a,b,c,d

  • Consider 2 possible solutions

    • Solution 1: (a,b) (c,d)

    • Solution 2: (c,d) (a,b)

  • They are the same kind of pairing but they are counted twice in the process (double counting)

    • Although in solution 1 (a,b) are paired up first while in solution 2 (c,d) are paired up first


Slide6 l.jpg

  • How about n = 3?

  • Suppose n = 3 and we have members a,b,c,d,e,f

  • How many “double counting” do we have for the same kind of pairing (a,b) (c,d) (e,f)?

    • Solution 1: (a,b) (c,d) (e,f)

    • Solution 2: (a,b) (e,f) (c,d)

    • Solution 3: (c,d) (a,b) (e,f)

    • Solution 4: (c,d) (e,f) (a,b)

    • Solution 5: (e,f) (a,b) (c,d)

    • Solution 6: (e,f) (c,d) (a,b)

  • Answer: 6 or 3!

  • 3! is the way of ordering the 3 pairs


Slide7 l.jpg

  • Generally, for 2n members, the counting process we introduced would produce n! double counting

    • n! is the way of ordering the n pairs

  • Our counting process counts

  • After eliminating the double counting, we have the solution to the tennis club problem


  • Login