Lecture 11 Binomial theorem

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Lecture 11 Binomial theorem

• June 25th , 2003

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• Pascals Triangle
• C(0,0)
• C(1,0) C(1,1)
• C(2,0) C(2,1) C(2,2)
• C(3,0) C(3,1) C(3,2) C(3,3)
• C(4,0) C(4,1) C(4,2) C(4,3) C(4,4)
• C(5,0) C(5,1) C(5,2) C(5,3) C(5,4)
  C(5,5)
• C(n,0) C(n,1)
  C(n,n-1) C(n,n)

0 1 2 3 4 5 n
Row
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• Pascal triangle (cont.)
• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• 1 6 15 20 15 6 1

C(n,k) C(n-1,k-1) C(n-1,k) 1ltklt(n-1)
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• Pascal triangle -- proof
• Mathematical proof
• Combinatorial proof.

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Binomial Theorem

• For every nonnegative integer n,

C(n,k) -- binomial coefficient
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Binomial Theorem--proof

• Mathematic Induction.
• Combinatorial methods

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Binomial Theorem-application

• (x-3)4C(4,0)x4(-3)0C(4,1)x3(-3)1C(4,2)x2(-3)2
• C(4,3)x1(-3)3C(4,4)x0(-3)4
• x4 - 12x3 54x2 - 108x 81
• The term k1 in the expansion of (ab)n is C(n,k)
  an-kbk.
• 2n C(n,0) C(n,1) C(n,k) C(n,n)

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Introduction to probability
S - Sample space E Event P(E)
Probability Example- Find the probability of
drawing an ace from a standard deck of cards.
S52, E4, P(E)4/52
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Probability distribution
Example ---practice 38
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Assignment 3

• Exercise 3.6
• 1b, 1f, 15, 18