Transcript Document

12
Counting
Just How Many Are There?
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section
Section12.3,
1.1, Slide
Slide11
12.3 Permutations and
Combinations
• Calculate the number of
permutations of n objects taken r
at a time.
• Use factorial notation to
represent the number of
permutations of a set of objects.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 2
12.3 Permutations and
Combinations
• Calculate the number of
combinations of n objects taken r
at a time.
• Apply the theory of permutations
and combinations to solve
counting problems.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 3
Permutations
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 4
Permutations
• Example: How many permutations are there of
the letters a, b, c, and d? Write the answer using
P(n, r) notation.
• Solution: We write the letters a, b, c, and d in a
line without repetition, so abcd and bcad are two
such permutations.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 5
Permutations
The slot diagram indicates there are
4 × 3 × 2 × 1 = 24 possibilities.
This can be written more succinctly as
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 6
Permutations
• Example: How many permutations are there of
the letters a, b, c, d, e, f, and g if we take the
letters three at a time? Write the answer using
P(n, r) notation.
• Solution: The slot
diagram indicates there
are 7 × 6 × 5 = 210
possibilities. This can be
written in permutation
notation as shown.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 7
Factorial Notation
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 8
Factorial Notation
• Example: Compute (8 – 3)!.
• Solution: We work inside parentheses first.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 9
Factorial Notation
• Example: Compute
.
• Solution:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 10
Factorial Notation
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 11
Factorial Notation
• Example: The 12-person theater group wishes
to select one person to direct a play, a second to
supervise the music, and a third to handle
publicity, tickets, and other administrative details.
In how many ways can the group fill these
positions?
• Solution: This is a permutation of selecting 3
people from 12.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 12
Combinations
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 13
Combinations
• Example: How many three-element sets can
be chosen from a set of five objects?
• Solution: Order is not important, so it is clear
that this is a combination problem.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 14
Combinations
• Example: How many four-person committees
can be formed from a set of 10 people?
• Solution: Order is not important, so it is clear
that this is a combination problem.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 15
Combinations
• Example: A syndicate has $15 million to spend
on tickets for a lottery. Tickets cost $1 and
contain a combination of six numbers from 1 to
44. Does the syndicate have enough money to
buy enough tickets to be guaranteed a winner?
• Solution: The combination of 6 numbers from
the 44 possible gives the number of different
tickets. That is, C(44, 6) = 7,059,052. Thus, the
syndicate has more than enough money to buy
enough tickets to be guaranteed a winner.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 16
Combinations
• Example: In the game of poker, five cards are
drawn from a standard 52-card deck. How many
different poker hands are possible?
• Solution:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 17
Combinations
• Example: In the game of bridge, a hand
consists of 13 cards drawn from a standard 52card deck. How many different bridge hands are
there?
• Solution:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 18
Combining Counting Methods
• Example: Two men and two women from a firm
will attend a conference. If the firm has ten men
and nine women, in how many different ways can
the conference attendees be selected?
• Solution: The answer is not C(19, 4) since this
includes options like four men and no woman
being sent to the conference.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 19
Combining Counting Methods
Stage 1: Select the two women from the nine
available.
Stage 2: Select the two men from the ten
available.
Thus, choosing the women and then choosing
the men can be done in
ways.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 20
Combining Counting Methods
• Example: A 16-member consortium wishes to
choose a committee consisting of a president, a
vice president, and a three-member executive
board. In how many different ways can this
committee be formed?
• Solution: We will count this in two stages:
(a) choosing the president and vice president
from the consortium, (b) choosing an executive
board from the remaining members.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 21
Combining Counting Methods
Stage 1: Choose the president and vice
president. This can be done in P(16, 2) ways.
Stage 2: Select the executive board. This can be
done in C(14, 3) ways.
Total:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 22
Combining Counting Methods
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 23
Combining Counting Methods
For example, consider the set {1, 2, 3, 4} and the 4th
row of Pascal’s triangle: 1 4 6 4 1.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 24
Combining Counting Methods
For example, consider the 4th row of Pascal’s
triangle: 1 4 6 4 1.
C(4, 0) = 1
C(4, 1) = 4
C(4, 2) = 6
C(4, 3) = 4
C(4, 4) = 1
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 25
Combining Counting Methods
• Example: Assume that a pharmaceutical
company has developed five antibiotics and four
immune system stimulators. In how many ways
can we choose a treatment program consisting of
three antibiotics and two immune system
stimulators to treat a disease? Use Pascal’s
triangle to speed your computations.
• Solution: We will count this in two stages:
(a) choosing the antibiotics, (b) choosing the
immune system simulators.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 26
Combining Counting Methods
Stage 1: Choosing 3 antibiotics from 5 can be
done in C(5, 3) ways.
Stage 2: Choosing 2 immune system simulators
from 4 can be done in C(4, 2) ways.
Total: C(5, 3) × C(4, 2) = 10 × 6 = 60 ways.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 12.3, Slide 27