ics141-lecture24-Combinatorics2

ics141-lecture24-Combinatorics2 - 24-1ICS 141: Discrete...

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Unformatted text preview: 24-1ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga24-2ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 24Chapter 5. Counting5.4 Binomial Coefficients5.5 Generalized Permutations and Combinations24-3ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSome material in these slides were taken/adapted from the slides made by Dr. Pomplun Some slides were done by Prof. Baek24-4ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiBinomial CoefficientsExpressions of the form C(n,r) are also called binomial coefficientsCoefficients of the expansion of powers of binomial expressions Binomial expression is a simply the sum of two terms such as x+yExample:32233223333)3,3()2,3()1,3(),3())(())()(()(yxyyxxyCxyCyxCxCyyyyyxyxyyxxxyyxyxxxyxxxyxyyyxxyxxyxyxyxyx+++=+++=+++++++=++++=+++=+24-5ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiThe Binomial TheoremLet xand ybe variables, and let nbe a nonnegative integer. ThenTo obtain a term of the form xn-jyj, it is necessary to choose n-jxs from the nsums, so that the other jterms in the product are ys. Therefore, the coefficient of xn-jyjis C(n, n-j) = C(n, j).The binomial theorem gives the coefficients of the expansion of powers of binomial expressions.( 29nnnnnjjnnjnynnxynnyxnyxnxnyxjnyx+-++++==+----=122112124-6ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiExamples(a+b)9 the coefficient of a5b4 = C(9,4)The coefficient of x12y13 in the expansion of (2x3y)25By binomial theoremThe coefficient of x12y13 is obtained when j= 13(x+y+z)9 the coefficient of x2y3z4= C(9,2)C(7,3)( 29jjjyxjyx)3()2(25)3(2252525-=-+-=1312131232!12!13!25)3(2)13,25(-=-C24-7ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiTProof:T Proof:It implies that:integer.positiveaiswhere,nknnkk)1(=-=Corollariesinteger.enonnegativaiswherenknnnk,2==-=-=-+====-knknnkknkkknnn)1()1(1))1(1(=-===+=nkkknnknnknkn11)11(2+++=+++53142nnnnnn24-8ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiTProof:Corollaries (cont.)integer.enonnegativaiswhere,nknnnkk32==knkkknnknnknkn221)21(3=-===+=24-9ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLet nand kbe positive integers with nk. ThenProofLet Xbe a set with nelements. Let a X....
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This note was uploaded on 09/12/2008 for the course ICS 141 taught by Professor Idk during the Fall '08 term at Hawaii.

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ics141-lecture24-Combinatorics2 - 24-1ICS 141: Discrete...

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