# 26 - Binomial Discrete Mathematics Andrei Bulatov Discrete...

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Introduction Binomial Coefficients Discrete Mathematics Andrei Bulatov

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Discrete Mathematics – Binomial Coefficients 26-2 Previous Lecture Combinations with repetitions C(n + r – 1,r – 1) Also recall that )! ( ! ! ) , ( r n r n r n C r n - = =
Discrete Mathematics – Binomial Coefficients 26-3 A Binomial A binomial is simply the sum of two terms, such as x + y We are to determine the expansion of Let us start with Every term in the expansion is obtained as the product of a term from the first binomial, a term from the second binomial, and a term from the third binomial Each of the terms xxy, xyx, and yxx is obtained by selecting y from one of the 3 binomials. Therefore, the coefficient 3 of is, actually, the number of 1-combinations from a set with 3 elements n y x ) ( + 3 y x ) ( + ) ( ) ( ) ( ) ( y x y x y x y x 3 + + + = + xyy xyx xxy yxx xyy xyx xxy xxx = 3 2 2 3 3 3 y xy y x x + + + = y x 2

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Discrete Mathematics – Binomial Coefficients 26-4 The Binomial Theorem Theorem . Let x and y be variables, and let n be a nonnegative integer. Then Proof. The terms in the product when it is expanded are of the form for j = 0,1,2,…, n. To count the number of terms of the form , note that to obtain such a term it is necessary to choose j y’s from the n binomials (so that the other n – j terms in the product are x’s).
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