Binomial Coefficients and Generating Functions - Cs.ioc.ee

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Counting elements of two sets Convolution Rule Let A(x) be the generating function for selecting items from set A , and let B(x), be the generating function for selecting items from set B. If A and B are disjoint, then the generating function for selecting items from the union A ∪ B is the product A(x) · B(x). Proof. To count the number of ways to select n items from A ∪ B we have to select j items from A and n − j items from B, where j ∈ {0,1,2,...,n}. Summing over all the possible values of j gives a total of a0bn + a1bn−1 + a2bn−2 + ··· + anb0 ways to select n items from A ∪ B. This is precisely the coefficient of x n in the series for A(x) · B(x) Q.E.D.
How many positive integer solutions does the equation x1 + x2 = n have? We accept any natural number can be solution for x1, i.e generating function for selection a value for this variable is A(x) = 1 + x + x2 + x3 + ··· = 1 1−x ; same for variable x2; all pairs of numbers for (x1,x2) are selected by the function H(x) = A(x) · A(x); numer of solutions is the coefficient of xn . H(x) = (1 + x + x 2 + x 3 + ···)(1 + x + x 2 + x 3 + ···) = = 1 · (1 + x + x 2 + x 3 + ···) + x(1 + x + x 2 + x 3 + ···) + +x 2 (1 + x + x 2 + x 3 + ···) + x 3 (1 + x + x 2 + x 3 + ···) + ··· = = 1 + 2x + 3x 2 + 4x 3 + ··· + (n + 1)x n 1 + ··· = (1 − x) 2 Indeed, this equation has n + 1 solutions: 0 + n = n 1 + (n − 1) = n 2 + (n − 2) = n ··· ··· n + 0 = n
Page 1 and 2: Binomial Coefficients and Generatin
Page 3 and 4: Next section 1 Binomial coefficient
Page 5 and 6: Binomial coefficients and combinati
Page 7 and 8: Properties of Binomial Coefficients
Page 9 and 10: Yet Another Proerty 4 If n,k > 0, t
Page 11 and 12: Pascal’s Triangle 1 1 1 1 2 1 1 3
Page 13 and 14: Next section 1 Binomial coefficient
Page 15 and 16: More examples (1) 〈1,1,1,1,...〉
Page 17 and 18: More examples (2) 〈a,ab,ab 2 ,ab
Page 19 and 20: More examples (3) Taking in the las
Page 21 and 22: Next subsection 1 Binomial coeffici
Page 23 and 24: Operations on Generating Functions
Page 35 and 36: Counting with Generating Functions
Page 37 and 38: Building Generating Functions that
Page 39 and 40: Building Generating Functions that
Page 41: Building Generating Functions that
Page 45 and 46: How many positive integer solutions
Page 47 and 48: Number of solutions of the equation
Page 49 and 50: Distribute n objects into k bins so
Page 51 and 52: Example: 100 Euro How many ways 100
Page 53 and 54: Example: 100 Euro (2) 1. Observatio
Page 55 and 56: Example: 100 Euro (3) Convolution:
Page 57 and 58: Vandermonde’s identity Binomial t
Page 59 and 60: Vandermonde’s identity (2) Equati
Page 61 and 62: Vandermonde’s identity (3) Specia
Page 63: Some other useful sequences For any

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