R' Johnsonbaugh Discrete Mathematics 5th edition, 2001

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R. JohnsonbaughDiscrete Mathematics
5th edition, 2001

• Chapter 4
• Counting methods
• and the pigeonhole principle

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4.1 Basic principles

• Multiplication principle
• If an activity can be performed in k successive
  steps,
• Step 1 can be done in n1 ways
• Step 2 can be done in n2 ways
• Step k can be done in nk ways
• Then the number of different ways that the
• activity can be performed is the product
• n1n2nk

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Addition principle

• Let X1, X2,, Xk be a collection of k pairwise
  disjoint sets, each of which has nj elements, 1 lt
  j lt k, then the union of those sets
• k
• X ? Xj
• j 1
• has n1 n2 nk elements

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4.2 Permutations and combinations

• A permutation of n distinct elements x1, x2,, xn
  is an ordering of the n elements. There are n!
  permutations of n elements.
• Example there are 3! 6 permutations of three
  elements a, b, c
• abc bac cab
• acb bca cba

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r-permutations

• An r-permutation of n distinct elements is an
  ordering of an r-element subset of the n elements
  x1, x2,, xn
• Theorem 4.2.10
• For r lt n the number of r-permutations of a set
  with n distinct objects is
• P(n,r) n(n-1)(n-2)(n-r1)

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Combinations

• Let X x1, x2,, xn be a set containing n
• distinct elements
• An r-combination of X is an unordered selection
  of r elements of X, for r lt n
• The number of r-combinations of X is the binomial
  coefficient
• C(n,r) n! / r!(n-r)! P(n,r)/ r!
  

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Generalized permutations

• Theorem 4.6.2
• Suppose that a sequence of n items has nj
  identical objects of type j, for 1lt j lt k.
• Then the number of orderings of S is
• ____n!____
• n1!n2!...nk!

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Generalized combinations

• Theorem 4.6.5 Suppose that X is a set
  containing t distinct elements.
• Then the number of unordered, k-element
  selections from X, repetitions allowed, is
• C(k t -1, t -1) C(k t -1, k)

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4.7 Binomial coefficients

• Theorem 4.7.1 Binomial theorem.
• For any real numbers a, b, and nonnegative
  integer n
• (ab)n C(n,0)anb0 C(n,1)an-1b1
• C(n,n-1)a1bn-1
  C(n,n)a0bn

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Pascals Triangle

• 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
• 1 7 21 35 35
  21 7 1
• 1 8 28 56 70
  56 28 8 1
• 1 9 36 84 126 126 84
  36 9 1

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4.8 The pigeonhole principle

• First form If k lt n and n pigeons fly into k
  pigeonholes, some pigeonhole contains at least
  two pigeons.

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Second form of the pigeonhole principle

• If X and Y are finite sets with X gt Y and f
  X ? Y is a function, then f(x1) f(x2) for some
  x1, x2 ? X, x1 ? x2.

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Third form of the pigeonhole principle

• If X and Y are finite sets with X n, Y m
  and k ?n/m?, then there are at least k values
  a1, a2,, ak ? X such that f(a1) f(a2)
  f(ak).
• Example
• n 5, m 3
• k ?n/m? ?5/3? 2.