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Unformatted text preview: 11COMP170 – Tutorial 2Combinatorial proofsVSAlgebraic onesRelationship betweenonetoone,onto,andinversefunctions21The problemConsider the identity:n2n24=n4n4222The problemConsider the identity:n2n24=n4n42Example:10284=45·70=3150=210·15=1046223The problemConsider the identity:n2n24=n4n42In the next slide we will see that is easy to prove thisalgebraicallyusing the formal algebraic definition of(nk).But,what does this proof mean?24The problemConsider the identity:n2n24=n4n42In the next slide we will see that is easy to prove thisalgebraicallyusing the formal algebraic definition of(nk).But,what does this proof mean?We will then see a combinatorial interpretation of the equation. This interpretation will provide a second,combinatorialproof.31n2n24=n!2!(n2)!(n2)!4!(n6)!=n!2! 4! (n6)!=n!4!(n4)!(n4)!2!(n6)!=n4n42An Algebraic Proof41Combinatorial ProofsAcombinatorialidentity is provenby counting some carefully chosen object in two different waysto obtain two different expressions of the same statement.51An exampleConsider the problem of choosing2 cochairmanand a4 person executive advisory boardfrom members of a 10personclub.52An exampleConsider the problem of choosing2 cochairmanand a4 person executive advisory boardfrom members of a 10personclub.There are(102)ways to choose the 2 cochairman and(84)waysof choosing the board from the remaining102=8people. Thisgives(102)(84).53An exampleConsider the problem of choosing2 cochairmanand a4 person executive advisory boardfrom members of a 10personclub.There are(102)ways to choose the 2 cochairman and(84)waysof choosing the board from the remaining102=8people. Thisgives(102)(84).Alternatively, we can first choose the 4 person board and thenthe 2 cochairman from the remaining104 = 6people. Thisgives(104)(62).54An exampleConsider the problem of choosing2 cochairmanand a4 person executive advisory boardfrom members of a 10personclub.There are(102)ways to choose the 2 cochairman and(84)waysof choosing the board from the remaining102=8people. Thisgives(102)(84).We have counted the same thing two ways....
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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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