Tutorial3 - 1-1COMP170 – Tutorial 3Pairing People...

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Unformatted text preview: 1-1COMP170 – Tutorial 3Pairing People UpAvoiding Double CountingOther Problems2-1The Tennis Club ProblemA tennis club has2nmembers.We want topair upthe members (by twos) to playsingles matches.2-2The 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?2-3The 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?3-1The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n-1members.3-2The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n-1members.Take the next remaining member(in the given order)andpair himwith any of the remaining2n-3members.3-3The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n-1members.Take the next remaining member(in the given order)andpair himwith any of the remaining2n-3members.Continue this way, at theithstep choosing the nextremaining member (in the order), thenpairing himwith any of the remaining2n-(2i+ 1)members.3-4The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n-1members.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 remaining2n-3members.Continue this way, at theithstep choosing the nextremaining member (in the order), thenpairing himwith any of the remaining2n-(2i+ 1)members.3-5The Tennis Club ProblemSort the members in some order.Take the first member andpair himwith any of the other2n-1members.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 remaining2n-3members.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.n-1Yi=0(2n-2i-1) = 1×3×...×(2n-1) =(2n)!n!2n.4-1The Tennis Club Problem (Con’t)In addition, we also determine who serves first foreach pairing. Now in how many ways can we specifyour pairs?4-2The 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.

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Tutorial3 - 1-1COMP170 – Tutorial 3Pairing People...

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