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COMP 170 L2 Page 1 L03: Binomial Coefficients l Purpose n Properties of binomial coefficients n Related issues: the Binomial Theorem and labeling

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COMP 170 L2 Page 2 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients

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COMP 170 L2 Page 3 Basic properties

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COMP 170 L2 Page 4 Basic Properties l Correct, but not so telling.

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COMP 170 L2 Page 5 Proof of.

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COMP 170 L2 Page 6 Proof of.

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COMP 170 L2 Page 7 Proof of.

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COMP 170 L2 Page 8 Basic Properties l Example

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COMP 170 L2 Page 9 Proof of

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COMP 170 L2 Page 10

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COMP 170 L2 Page 11 Proof of

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COMP 170 L2 Page 12 Proof of

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COMP 170 L2 Page 13 Summary of Basic Properties

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COMP 170 L2 Page 14 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients

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COMP 170 L2 Page 15 Pascal’s Triangle

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COMP 170 L2 Page 16 Pascal’s Triangle l Each entry = sum of the two entries above it

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COMP 170 L2 Page 17 Pascal’s Triangle l Each entry = sum of the two entries above it l Next row?

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COMP 170 L2 Page 18 Pascal Relationship l Examples

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COMP 170 L2 Page 19 Algebraic Proof of Pascal’s Relationship l For reference only. l Will give proof by sum principle. More revealing.

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COMP 170 L2 Page 20 Proof of Pascal’s Relationship by Sum Principle

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COMP 170 L2 Page 21 Proof of Pascal’s Relationship by Sum Principle

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COMP 170 L2 Page 22

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COMP 170 L2 Page 23 Pascal Relationship

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COMP 170 L2 Page 24 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients

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COMP 170 L2 Page 25 Expanding Binomials

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COMP 170 L2 Page 26 The Binomial Theorem l We are concerned with n What is the theorem true?

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COMP 170 L2 Page 27 Examples l Monomial terms: n Lists of length two, each element can either be x or y. l How many monomial terms with one y (and hence one x) ? n = number of ways to choose 1 place among 2 places n That is the coefficient for the term l Similarly n Coefficient for = number of lists having 0 place for y = n Coefficient for = number of lists having 2 places for y = l So

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COMP 170 L2 Page 28 Examples l Coefficient for n = number of ways to choose 2 places for 3 places. l Coefficient for n = number of ways to choose i places from 3 places

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COMP 170 L2 Page 29 Proof of the Binomial Theorem l Coefficient of n = number of lists having y in k places n =number of ways to choose k places from n places n=n=

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COMP 170 L2 Page 30 Applications of the Binomial Theorem

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COMP 170 L2 Page 31 Applications of the Binomial Theorem

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COMP 170 L2 Page 32 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients

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COMP 170 L2 Page 33 Labeling with 2 Colors

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COMP 170 L2 Page 34 Labeling with 3 Colors

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COMP 170 L2 Page 35 Trinomial Coefficients

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COMP 170 L2 Page 36 Number of Partitions

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COMP 170 L2 Page 37 Trinomial Coefficients The number of ways to partition a set of n places into 3 subsets of k1, k2 and k3 places Each list is of length n, consisting of x, y, z

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COMP 170 L2 Page : Recap

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COMP 170 L2 Page : Recap

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COMP 170 L2 Page 40 Past Exam Question

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COMP 170 L2 Page 41 Past Exam Question

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