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Unformatted text preview: 241ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga242ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 24Chapter 5. Counting5.4 Binomial Coefficients5.5 Generalized Permutations and Combinations243ICS 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. Baek244ICS 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+++=+++=+++++++=++++=+++=+245ICS 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 xnjyj, it is necessary to choose njxs from the nsums, so that the other jterms in the product are ys. Therefore, the coefficient of xnjyjis C(n, nj) = C(n, j).The binomial theorem gives the coefficients of the expansion of powers of binomial expressions.( 29nnnnnjjnnjnynnxynnyxnyxnxnyxjnyx+++++==+=1221121246ICS 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(=C247ICS 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+++=+++53142nnnnnn248ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiTProof:Corollaries (cont.)integer.enonnegativaiswhere,nknnnkk32==knkkknnknnknkn221)21(3====+=249ICS 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.
 Fall '08
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