Page 1 COMP 170 L2 L03: Binomial Coefficients Purpose Properties of binomial coefficients Related issues: the Binomial Theorem and labeling
COMP 170 L2 Page 1 L03: Binomial Coefficients l Purpose n Properties of binomial coefficients n Related issues: the Binomial Theorem and labeling.
Dec 20, 2015
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Page 1COMP 170 L2
L03: Binomial Coefficients
Purpose Properties of binomial coefficients
Related issues: the Binomial Theorem and
labeling
Page 2COMP 170 L2
Outline
Basic properties
Pascal’s triangle
The Binomial theorem
Labeling and Trinomial coefficients
Page 3COMP 170 L2
Basic properties
Page 4COMP 170 L2
Basic Properties
Correct, but not so telling.
Page 5COMP 170 L2
Proof of .
Page 6COMP 170 L2
Proof of .
Page 7COMP 170 L2
Proof of .
Page 8COMP 170 L2
Basic Properties
Example
Page 9COMP 170 L2
Proof of
Page 10COMP 170 L2
Page 11COMP 170 L2
Proof of
Page 12COMP 170 L2
Proof of
Page 13COMP 170 L2
Summary of Basic Properties
Page 14COMP 170 L2
Outline
Basic properties
Pascal’s triangle
The Binomial theorem
Labeling and Trinomial coefficients
Page 15COMP 170 L2
Pascal’s Triangle
Page 16COMP 170 L2
Pascal’s Triangle
Each entry = sum of the two entries above it
Page 17COMP 170 L2
Pascal’s Triangle
Each entry = sum of the two entries above it
Next row?
Page 18COMP 170 L2
Pascal Relationship
Examples
Page 19COMP 170 L2
Algebraic Proof of Pascal’s Relationship
For reference only. Will give proof by sum principle. More revealing.
Page 20COMP 170 L2
Proof of Pascal’s Relationship by Sum Principle
Page 21COMP 170 L2
Proof of Pascal’s Relationship by Sum Principle
Page 22COMP 170 L2
Page 23COMP 170 L2
Pascal Relationship
Page 24COMP 170 L2
Outline
Basic properties
Pascal’s triangle
The Binomial theorem
Labeling and Trinomial coefficients
Page 25COMP 170 L2
Expanding Binomials
Page 26COMP 170 L2
The Binomial Theorem
We are concerned with What is the theorem true?
Page 27COMP 170 L2
Examples
Monomial terms: Lists of length two, each element can either be x or y.
How many monomial terms with one y (and hence one x) ?
= number of ways to choose 1 place among 2 places That is the coefficient for the term
Similarly Coefficient for
= number of lists having 0 place for y = Coefficient for
= number of lists having 2 places for y =
So
Page 28COMP 170 L2
Examples
Coefficient for = number of ways to choose 2 places for 3 places.
Coefficient for = number of ways to choose i places from 3 places
Page 29COMP 170 L2
Proof of the Binomial Theorem
Coefficient of = number of lists having y in k places
=number of ways to choose k places from n places
=
Page 30COMP 170 L2
Applications of the Binomial Theorem
Page 31COMP 170 L2
Applications of the Binomial Theorem
Page 32COMP 170 L2
Outline
Basic properties
Pascal’s triangle
The Binomial theorem
Labeling and Trinomial coefficients
Page 33COMP 170 L2
Labeling with 2 Colors
Page 34COMP 170 L2
Labeling with 3 Colors
Page 35COMP 170 L2
Trinomial Coefficients
Page 36COMP 170 L2
Number of Partitions
Page 37COMP 170 L2
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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