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Combinatorics

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Unformatted text preview: ﬃcients Generaliza5ons –  Combina5ons with repe55ons, permuta5ons with indis5nguishable objects •  Algorithms –  Genera5ng combina5ons (1), permuta5ons (2) •  More Examples CSCE 235 Combinatorics 42 Binomial Coeﬃcients (1) •  The number of r- combina5ons is also called the binomial coeﬃcient •  The binomial coeﬃcients are the coeﬃcients in the expansion of the expression, (mul5variate polynomial), (x+y)n •  A binomial is a sum of two terms CSCE 235 Combinatorics 43 Binomial Coeﬃcients (2) •  Theorem: Binomial Theorem Let x, y, be variables and let n be a nonnega5ve integer. Then Expanding the summa5on we have Example CSCE 235 Combinatorics 44 Binomial Coeﬃcients: Example •  What is the coeﬃcient of the term x8y12 in the expansion of (3x+4y)20? –  By the binomial theorem, we have –...
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