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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?.

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the tennis club problem supplementary notes
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

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

How many ways can we perform this process?

  • By the product principle, the number of ways is
slide4

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

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

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

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
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