comp 170 l2 page 1 l03: binomial coefficients l purpose n properties of binomial coefficients n...
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Page 1COMP 170 L2
L03: Binomial Coefficients
Purpose Properties of binomial coefficients
Related issues: the Binomial Theorem and
labeling
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Outline
Basic properties
Pascal’s triangle
The Binomial theorem
Labeling and Trinomial coefficients
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Basic properties
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Basic Properties
Correct, but not so telling.
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Proof of .
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Proof of .
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Proof of .
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Basic Properties
Example
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Proof of
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Proof of
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Proof of
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Summary of Basic Properties
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Outline
Basic properties
Pascal’s triangle
The Binomial theorem
Labeling and Trinomial coefficients
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Pascal’s Triangle
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Pascal’s Triangle
Each entry = sum of the two entries above it
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Pascal’s Triangle
Each entry = sum of the two entries above it
Next row?
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Pascal Relationship
Examples
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Algebraic Proof of Pascal’s Relationship
For reference only. Will give proof by sum principle. More revealing.
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Proof of Pascal’s Relationship by Sum Principle
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Proof of Pascal’s Relationship by Sum Principle
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Pascal Relationship
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Outline
Basic properties
Pascal’s triangle
The Binomial theorem
Labeling and Trinomial coefficients
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Expanding Binomials
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The Binomial Theorem
We are concerned with What is the theorem true?
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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
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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
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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
=
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Applications of the Binomial Theorem
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Applications of the Binomial Theorem
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Outline
Basic properties
Pascal’s triangle
The Binomial theorem
Labeling and Trinomial coefficients
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Labeling with 2 Colors
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Labeling with 3 Colors
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Trinomial Coefficients
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Number of Partitions
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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