Counting

1
Chapter 5

• Counting

2
Sec 5.1

• The Basics of Counting

3
Basic Counting Principles

• The Product Rule If an experiment can be
  modeled as k sequential steps, where the first
  step can be done in n1 ways the second in n2
  and so forth with the kth step done in nk ways,
  then the total number of possible outcomes for
  the experiment is n1n2n3nk.
• Since there is an intrinsic order (1st, 2nd, 3rd,
  ) in the steps forming the experiment, the total
  number of outcomes can be modeled as the number
  of ordered n-tuples in the set A1xA2xA3xxAk
  where A1 has n1 element, A2 has n2, , Ak has
  elements nk elements.

4
Basic Counting Principles

• The Sum Rule If the outcomes for an experiment
  can be modeled as elements in k distinct disjoint
  (non- overlapping) sets, A1, A2, A3, , Ak then
  the total number of possible outcomes for the
  experiment isA1?A2?A3 ? ?Ak A1 A2
  A3 Ak.
• When using the sum rule there should be no
  significance to the ordering of the elements in
  the outcome for the experiment.

5
Counting Rules

• The Inclusion-Exclusion Principle If the
  outcomes for an experiment can be modeled as
  elements in the union of two sets, A and B, then
  the number of possible outcomes for the
  experiment is A?B A B - A?B
• The Complement Principle If an experiment has n
  possible outcomes, then the number of ways that a
  particular set A of outcomes cannot occur is n -
  A.

6
Homework

• pg. 344 1, 3, 4, 9, 10, 11, 19, 21, 23, 27, 33,
  41

7
Sec 5.2

• The Pigeonhole Principle

8
The Pigeonhole Principle

• If K1 or more object are placed into K boxes,
  then at least one box contains two or more
  objects
• The Generalized Pigeonhole Principle
• If N objects are placed in k boxes, then there is
  at least one box containing ?N/k? objects.

9
Examples

• Every sequence of n2 1 distinct real numbers
  contains a subsequence of length n 1 that is
  either strictly increasing or strictly
  decreasing.
• Ramsey Numbers Let m and n be positive
  integers, than R(m,n) denotes the minimum number
  of people at a party such that there are either m
  mutual friends or n mutual enemies, then R(3,3)
  6

10
Homework

• pg. 353 3, 5, 9, 13, 15, 17, 19, 31, 33

11
Sec 5.3

• Permutations and Combinations

12
Definitons

• An r-permutation of n distinct objects is a order
  arrangement of these r elements. The number of
  such rearrangements is P(n,r) n(n-1)(n-2)
  (n-r1)
• An r-combination of the n distinct elements of a
  set is an unordered selection of r elements from
  this set. (A subset of size r). The number of
  such combinations is C(n,r) n!/(r!(n-r)!)
• Theorem C(n,r) C(n,n-r)

13
Homework

• pg. 360 3, 7, 11, 13, 19, 21a,b,c,d, 23, 27,
  33
• End Chapter 5