Transcript Chapter 7 Part Two Counting Techniques

Chapter 7 - Part Two
Counting Techniques
Wednesday, March 18, 2009
Permutations:
 A permutation of r elements from a set of n
elements in any specific ordering or arrangement,
without repetition of the r elements. Each
arrangement is a different permutation.
 Clue words: arrangement, schedule, order,....
 Example: There are six permutations of the
letters A, B, and C.
 ABC ACB BAC BCA CBA CAB
Permutations Formula
n!
P(n, r )  n Pr 
(n  r )!
Example
In the Olympics Gymnastics competition,
8 gymnasts compete for medals. How
many ways can the medals be awarded
(gold, silver, and bronze)?
Distinguishable Permutations
 Objects are not all distinguishable, namely n1 of
type 1, n2 of type 2, etc. The number of
permutations is:
n!
n1!n2!  nm !
Example
How many permutations are there of the
letters in the word STATISTICS?
Combinations
A subset of items selected without
regard to order.
Clue words: group, committee, sample....
Example: There is only one combination of
the letters A, B, and C === ABC
Combinations Formula
 n
n!
C (n, r )  n Cr    
 r  (n  r )!r!
Example
How many committees of three people can
be selected from a group of 8 people?
Pascal’s Triangle
Can be used to compute combinations
Baseball
 How many ways can three outfielders
and four infielders be chosen from five
outfielders and seven infielders?
Lottery
 In the Pennsylvania lottery drawing, 5
numbered balls are selected from a box
containing balls numbered 1 through 40.
How many different combinations of
winning numbers are there?
Some Card Problems
I am playing a hand of 5 card poker.
What is the probability that I am dealt the
following:
 3 Kings and 2 Aces?
 All hearts
 Exactly two aces
 Three of a kind.
Example
A barrel contains 15 apples. Of the
apples, 5 are rotten and 10 are good.
Three apples are selected at random.
What is the probability of selecting at least
one good apple?
Binomial Probability
 Same experiment is repeated several
times.
 Only two possible outcomes: success
and failure
 Repeated trials are independent.
 n = number of trials
 x = number of successes
 p = probability of success on each trial
Formula
 n x
n x
P( x)    p (1  p)
x
 
Example
Flip a coin 20 times. What is the
probability of getting 6 tails?
Example
I am taking a 10 question, multiple choice
exam and I have not studied. Each
question has 4 possible answers. By
guessing only, what is the probability that
I can get 6 questions correct?
Problem of the Day
A customer walks into a hardware store to
buy something and asks the clerk how
much 1 would cost and the clerk answers
$1. The customer then asks how much 10
would cost and the clerk answers $2. The
customer says, “I’ll buy 1515,” and pays
the clerk $4. What was the customer
buying?