Tutorial2 - 1-1COMP170 – Tutorial 2Combinatorial...

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Unformatted text preview: 1-1COMP170 – Tutorial 2Combinatorial proofsVSAlgebraic onesRelationship betweenone-to-one,onto,andinversefunctions2-1The problemConsider the identity:n2n-24=n4n-422-2The problemConsider the identity:n2n-24=n4n-42Example:10284=45·70=3150=210·15=104622-3The problemConsider the identity:n2n-24=n4n-42In the next slide we will see that is easy to prove thisalgebraicallyusing the formal algebraic definition of(nk).But,what does this proof mean?2-4The problemConsider the identity:n2n-24=n4n-42In 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 equa-tion. This interpretation will provide a second,combinato-rialproof.3-1n2n-24=n!2!(n-2)!(n-2)!4!(n-6)!=n!2! 4! (n-6)!=n!4!(n-4)!(n-4)!2!(n-6)!=n4n-42An Algebraic Proof4-1Combinatorial ProofsAcombinatorialidentity is provenby counting some care-fully chosen object in two different waysto obtain two dif-ferent expressions of the same statement.5-1An exampleConsider the problem of choosing2 co-chairmanand a4 person executive advisory boardfrom members of a 10-personclub.5-2An exampleConsider the problem of choosing2 co-chairmanand a4 person executive advisory boardfrom members of a 10-personclub.There are(102)ways to choose the 2 co-chairman and(84)waysof choosing the board from the remaining10-2=8people. Thisgives(102)(84).5-3An exampleConsider the problem of choosing2 co-chairmanand a4 person executive advisory boardfrom members of a 10-personclub.There are(102)ways to choose the 2 co-chairman and(84)waysof choosing the board from the remaining10-2=8people. Thisgives(102)(84).Alternatively, we can first choose the 4 person board and thenthe 2 co-chairman from the remaining10-4 = 6people. Thisgives(104)(62).5-4An exampleConsider the problem of choosing2 co-chairmanand a4 person executive advisory boardfrom members of a 10-personclub.There are(102)ways to choose the 2 co-chairman and(84)waysof choosing the board from the remaining10-2=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.

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Tutorial2 - 1-1COMP170 – Tutorial 2Combinatorial...

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