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Unformatted text preview: 11COMP170 – Tutorial 3Pairing People UpAvoiding Double CountingOther Problems21The Tennis Club ProblemA tennis club has2nmembers.We want topair upthe members (by twos) to playsingles matches.22The Tennis Club ProblemA tennis club has2nmembers.We want topair upthe members (by twos) to playsingles matches.(a) In how many ways can wepair upall the members of the club?23The Tennis Club ProblemA tennis club has2nmembers.We want topair upthe members (by twos) to playsingles matches.(b) Suppose that in addition to specifying whopairs upwith whom, we also determine whoserves firstfor each pairing.How many ways are there now to specify our pairs?(a) In how many ways can wepair upall the members of the club?31The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n1members.32The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n1members.Take the next remaining member(in the given order)andpair himwith any of the remaining2n3members.33The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n1members.Take the next remaining member(in the given order)andpair himwith any of the remaining2n3members.Continue this way, at theithstep choosing the nextremaining member (in the order), thenpairing himwith any of the remaining2n(2i+ 1)members.34The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n1members.Note that every possible way of pairing the2nis generatedexactly once by this procedure so it counts everythingTake the next remaining member(in the given order)andpair himwith any of the remaining2n3members.Continue this way, at theithstep choosing the nextremaining member (in the order), thenpairing himwith any of the remaining2n(2i+ 1)members.35The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n1members.Note that every possible way of pairing the2nis generatedexactly once by this procedure so it counts everythingTake the next remaining member(in the given order)andpair himwith any of the remaining2n3members.By the product principle, the number of ways isContinue this way, at theithstep choosing the nextremaining member (in the order), thenpairing himwith any of the remaining2n(2i+ 1)members.n1Yi=0(2n2i1) = 1×3×...×(2n1) =(2n)!n!2n.41The Tennis Club Problem (Con’t)In addition, we also determine who serves first foreach pairing. Now in how many ways can we specifyour pairs?42The Tennis Club Problem (Con’t)In addition, we also determine who serves first foreach pairing. Now in how many ways can we specifyour pairs?...
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This note was uploaded on 08/25/2010 for the course COMP COMP170 taught by Professor M.j.golin during the Spring '10 term at HKUST.
 Spring '10
 M.J.Golin
 Computer Science

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