L03_BinomialCoefficients

L03_BinomialCoefficients - 1-1COMP170Discrete Mathematical...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 1.3, pp. 19-26Binomial CoefficientsVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen2-11.3 Binomial Coefficients2-21.3 Binomial Coefficients•Pascal’s Triangle2-31.3 Binomial Coefficients•Pascal’s Triangle•A Proof using the Sum Principle2-41.3 Binomial Coefficients•Pascal’s Triangle•A Proof using the Sum Principle•The Binomial Theorem2-51.3 Binomial Coefficients•Pascal’s Triangle•A Proof using the Sum Principle•The Binomial Theorem•Labeling and Trinomial Coefficients3-1Some properties of Binomial Coefficients•nk=n!k!(n-k)!•n= 1only one set of size.•nn= 1only one set of sizen.•nk=nn-kObvious from equation. Can youthink of a simple bijection that explains this?is the number ofk-elementsubsets of ann-element set.4-1Some properties of Binomial Coefficients (cont)4-2Some properties of Binomial Coefficients (cont)nXi=0ni= 2n4-3Some properties of Binomial Coefficients (cont)nXi=0ni= 2nUse Sum PrincipleLetP=set of all subsets of{1,2,. . . ,n}Si=set of allisubsets of{1,2,. . . ,n}4-4Some properties of Binomial Coefficients (cont)nXi=0ni= 2nUse Sum PrincipleLetP=set of all subsets of{1,2,. . . ,n}Si=set of allisubsets of{1,2,. . . ,n}⇒|P|=nXi=0|Si|=nXi=0ni4-5Some properties of Binomial Coefficients (cont)nXi=0ni= 2nUse Sum PrincipleLetP=set of all subsets of{1,2,. . . ,n}Si=set of allisubsets of{1,2,. . . ,n}⇒|P|=nXi=0|Si|=nXi=0niLetL=L1L2...Lnbe a list of sizenfrom{,1}IfL=set of all such lists⇒|L|= 2nThere is abijection(next page)betweenLandPso|P|= 2nand we are done.5-1LetP=set of all subsets of{1,2,. . . ,n}LetL=L1L2...Lnbe a list of sizenfrom{,1}andL=set of all such lists5-2LetP=set of all subsets of{1,2,. . . ,n}LetL=L1L2...Lnbe a list of sizenfrom{,1}andL=set of all such listsDefine the following functionf:L →PIfL∈ Lthenf(L)is the setS⊆ {1,,2,...,n}defined byi∈S⇔Li= 15-3LetP=set of all subsets of{1,2,. . . ,n}LetL=L1L2...Lnbe a list of sizenfrom{,1}andL=set of all such listsDefine the following functionf:L →PIfL∈ Lthenf(L)is the setS⊆ {1,,2,...,n}defined byi∈S⇔Li= 1fis abijectionbetweenLandP(why?) so|L|=|P|5-4LetP=set of all subsets of{1,2,. . . ,n}LetL=L1L2...Lnbe a list of sizenfrom{,1}andL=set of all such listsDefine the following functionf:L →PIfL∈ Lthenf(L)is the setS⊆ {1,,2,...,n}defined byi∈S⇔Li= 1f(10101) ={1,3,5}, f(11101) ={1,2,3,5}, f(00000) =∅.Ex:n= 5fis abijectionbetweenLandP(why?) so|L|=|P|5-5LetP=set of all subsets of{1,2,. . . ,n}LetL=L1L2...Lnbe a list of sizenfrom{,1}andL=set of all such listsDefine the following functionf:L →PIfL∈ Lthenf(L)is the setS⊆ {1,,2,...,n}defined byi∈S⇔Li= 1f(10101) ={1,3,5}, f(11101) ={1,2,3,5}, f(00000) =∅....
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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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L03_BinomialCoefficients - 1-1COMP170Discrete Mathematical...

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