Fall 2014 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of...
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Transcript of Fall 2014 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of...
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Fall 2014COMP 2300 Discrete Structures for Computation
Donghyun (David) KimDepartment of Mathematics and Computer ScienceNorth Carolina Central University
Chapter 9.7Pascal’s Formula and the Binomial Theorem
2
Pascal’s Formula•
• Pascal’s formula: Let n and r be positive inte-gers and suppose . Then,
why?
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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3
Pascal’s Formula – cont’•
• Pascal’s formula: Let n and r be positive inte-gers and suppose . Then,
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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Pascal’sTriangle
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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Deriving New Formulas from Pascal’s Formula• Use Pascal’s formula to derive a formula for
in
terms of values of , , and . Assume n and
r are nonnegative integers and .
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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Deriving New Formulas from Pascal’s Formula – cont’• Use Pascal’s formula to derive a formula for
in
terms of values of , , and . Assume n and
r are nonnegative integers and .
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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The Binomial Theorem• In algebra a sum of two terms, such
as , is called a binomial.
• The binomial theorem gives an expression for the powers of a binomial , for each positive integer n and all real numbers a and b.
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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The Binomial Theorem – cont’•
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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The Binomial Theorem – cont’• Proof of the Binomial Theorem (By Math. In-
duction)
• When n=1,
• Suppose
Then, we have
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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Deriving Another Combinatorial Identity from the Binomial Theorem• Use the binomial theorem to show that
for all integers
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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Deriving Another Combinatorial Identity from the Binomial Theorem• Use the binomial theorem to show that
for all integers
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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Using a Combinatorial Argument to Derive the Identity• Show that
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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Using a Combinatorial Argument to Derive the Identity – cont’• Show that
• Suppose S is a set with n elements.
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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n
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Number of subsets of
S
Number of subsets of
size 0
Number of subsets of
size 1
Number of subsets of
size n
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14
Using a Combinatorial Argument to Derive the Identity – cont’• Show that
• Suppose S is a set with n elements.
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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Number of subsets of
S
Number of subsets of
size 0
Number of subsets of
size 1
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size n
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Using the Binomial Theorem to Sim-plify a Sum• Express the following sum in closed form
(without using a summation symbol and without using an ellipsis…):
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
n
k
k
k
n
0
9
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Using the Binomial Theorem to Sim-plify a Sum – cont’• Express the following sum in closed form
(without using a summation symbol and without using an ellipsis…):
•
Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University
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