Algorithms and Discrete Mathematics 20082009

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Algorithms and Discrete Mathematics 2008/2009

• Lecture 3
• Binomial coefficients

Ioannis Ivrissimtzis
27-Oct-2008
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Overview

• Summary of previous lecture
• Pascals triangle
• Pascals identity
• Binomial theorem

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Definitions

• The factorial of an integer n 0 is defined by
• and denoted by n!

An ordered arrangement of r elements of a set is
called an r-permutation.
An r-combination of elements of a set is an
unordered selection of r elements from the set.
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Formulae

• If n and r are integers with 1 r n, then
  there are
• r-permutations of a set with n distinct elements.

The number of r-combinations of a set with n
elements, where n and r are integers with 0 r
n, equals
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Overview

• Summary of previous lecture
• Pascals triangle
• Pascals identity
• Binomial theorem

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Pascals triangle
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Pascals triangle
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Pascals triangle

• Pascals triangle is an arrangement of the
  binomial coefficients in a
• triangular array called Pascals triangle.
• The nth row in the triangle consists of the
  binomial coefficients
• When two adjacent binomial coefficients in this
  triangle are added, the
• binomial coefficient in the next row between them
  is produced.

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Pascals triangle

• Pascals triangle is named after the French
  mathematician Blaise
• Pascal (1623-1662) who studied, among other
  things, combinatorial
• problems and probabilities.
• Pascals triangle was known long time before
  Pascal. Sometimes it is
• referred as
• Tartaglias triangle (Italy)
• Yanghuis triangle (China)
• Khayyams triangle (Iran)

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Pascals triangle
Arithmetica Integra Michael Stifel, 1544
Precious Mirror of the Four Elements Chu Shih-Chie
h, 1303
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Overview

• Summary of previous lecture
• Pascals triangle
• Pascals identity
• Binomial theorem

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Pascals identity
Pascals triangle is based on Pascals identity
for binomial coefficients.

• Pascals Identity Rosen, p.366 Let n and k be
  positive integers with
• n k. Then

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Pascals identity

• Proof

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Overview

• Summary of previous lecture
• Pascals triangle
• Pascals identity
• Binomial theorem

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Binomial theorem

• The Binomial Theorem Rosen, p.363 Let x and y
  be variables, and
• let n be a nonnegative integer. Then

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Binomial theorem

• Example 3.1

Example 3.2
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Binomial theorem

• Example 3.3 Rosen, p.364 Expand (xy)4

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Binomial theorem

• Exercise 3.4 Rosen, p.364 Let n be a
  nonnegative integer. Then

Proof Using the Binomial Theorem with x1 and
y1, we see that