The Tennis Club Problem Supplementary Notes

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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
• 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?
• 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

How many ways can we perform this process?

• By the product principle, the number of ways is

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

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

• 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)