Combinatorics

1
Combinatorics

• Multiplication Principle
• Permutations
• Addition Principle

2
Clowns and Clothes

• Complete the activity
• What do you notice about your guess and the
  number of different possibilities you were given?

3
The Multiplication Principle

• Let ni represent the number of ways of completing
  the ith step in a process of k steps.
• The number of ways to complete the process is
  given by the product n1 n2 n3 nk

4
Permutations

• The total number of ordered arrangements of a set
  of n distinct elements is given by the product
  n(n-1)(n-2).21
• This product can also be described as n! or n
  factorial

5
Permutations

• An ordered arrangement of a subset of r elements
  chosen from a set of n objects is called an
  r-permutation of the set
• The number of such r-permutations from a set of n
  object can be written as

6
License Plates

• Suppose that the license plates for a certain
  state display 3 letters followed by three digits
  (Example MFT-986).
• How many different license plates can be
  manufactured if no repetitions of letters or
  digits are allowed?
• The number of license plates is given by

7
High Museum

• Suppose that the Louvre has agreed to loan 19
  different paintings to the High Museum of Art as
  part of the new exhibit. However the exhibit
  space only has enough places to hang 8 paintings
  at a time. How many different arrangements of
  the paintings can be made?
• The answer can be determined by P(19, 8).

8
Music Appreciation

• To fulfill requirements for his Music
  Appreciation course, Raphael must attend one
  musical performance from among 3 voice recitals,
  4 piano recitals, and 2 jazz concerts.
• In how many different ways can Raphael meet the
  course requirement?
• The answer is 9.

9
The Addition Principle

• Suppose that X1, X2 Xk is a collection of
  disjoint sets and Xi has ni elements
• If a process is to be completed by choosing one
  element from exactly one of the sets in the
  collection, then the number of ways to complete
  the process is the sum n1 n2 n1 nk

10
Work on 6.1 Exploratory Exercise 5
Finish this Exercise for Wednesday to turn in.