Transcript Permutations and Combinations

OBJECTIVES
I will :
•
Compare and contrast permutations and combinations.
•
Understand terminology and variables associated with permutations and
combinations.
•
Identify and apply permutations and combinations in problem situations.
•
Use technology to compute probabilities of permutations and combinations.
FIRST A REVIEW OF THE MULTIPLICATION RULE
If there are n possible outcomes for event A,
and m possible outcomes for event B,
then there are n x m possible outcomes for event A followed by event B.
# of Outcomes = n x m
.
TOSS COIN, ROLL DICE EXAMPLE
Toss Coin
1
2
3
Head
Tail
Roll Dice
Roll Dice
4
5
6
1
2
3
4
Resulting Sample Space
H1, H2, H3, H4, H5, H6
T1, T2, T3, T4, T5, T6
5
6
MULTIPLICATION RULE
If there are n possible outcomes for event A and m possible outcomes for event B,
then there are n x m possible outcomes for event A followed by event B.
In our Coin Toss, Die Roll example, there are 2 possible outcomes for the Coin Roll (H
or T) and 6 possible events for the Die Toss (1,2,3,4,5, or 6).
There are 2 x 6 or 12 possible outcomes in the sample
space.
MULTIPLICATION RULE MORE EXAMPLES
Ford’s new Fusion comes with two body styles, three interior package options, and
four colors, as well as a choice of standard or automatic transmissions. The
dealership wants to carry one of each type in its inventory. How many Ford
Fusions will they order?
There are 2 body styles, 3 interior package options, 4 colors,
and 2 transmissions. There are 2 x 3 x 4 x 2 = 48 possible
outcomes.
MULTIPLICATION RULE MORE EXAMPLES
The Old Orange Café offers a special lunch menu each day including two appetizers,
three main courses, and four desserts. Customers can choose one dish from
each category. How many different meals can be ordered from the lunch menu?
There are 2 appetizers, 3 main courses, and 4 desserts.
There are 2 x 3 x 4 = 24 possible outcomes.
REVIEW FACTORIAL NOTATION
Remember, for a counting number, n
n! = n (n-1) (n-2) (n-3) …1 i.e., 5! = 5 x 4 x 3 x 2 x 1
0! = 1 (By special definition)
1! = 1
COMBINATIONS
•
The number of ways choices can be combined without repetition.
•
Order does not matter.
For example, how many ways can these four members of the tennis team be
combined to play a doubles match?
COMBINATIONS
How many ways can these four members of the tennis team be combined to play a
doubles match? The multiplication rule says there are 4 possible choices for the
1st team member x 3 possible choices for the 2 nd team member. Or 4 x 3 = 12
possible choices.
However, a quick illustration shows this is incorrect. There are in fact only 6 possible
unique combinations.
1
3
2
4
5
6
COUNTING RULE FOR COMBINATIONS
The number of combinations of n distinct objects ,
taking them r at a time, is
nCr
or Cn,r =
n!
r!(n-r)!
Where n and r are whole numbers.
n≥r
and n! (“n factorial”) = n x (n – 1) x (n – 2) x …. 1
COMBINATIONS
Using the combination formula nCr or Cn,r in our tennis team example, we calculate:
The number of combinations of 4 distinct players , taken 2 at a time, is
4C2
or C4,2 =
4!
=
2!(4-2)!
1
3
2
4!___ = 4 x 3 x 2 x 1 = 6
2!(2)!
4
2x1x2x1
5
6
CONSIDER ANOTHER SAMPLE SPACE
How many ways can these five members of the tennis team be combined to play a
doubles match?
Using the combination formula nCr or Cn,r we calculate:
The number of combinations of 5 distinct players , taken 2 at a
time, is
5!
= 5!___ = 5 x 4 x 3 x 2 x 1 = 10
5C2 or C5,2 =
2!(5 - 2)! 2!(3)!
2x1x3x2x1
AND ANOTHER SAMPLE SPACE
How many ways can these six members of the track team be combined to run a 4 x 4
relay?
Using the combination formula nCr or Cn,r we calculate:
The number of combinations of 6 distinct runners , taken 4 at a
time, is
6!
= 6!___ = 6 x 5 x 4 x 3 x 2 x 1 = 15
6C4 or C6,4 =
4!(6 - 4)! 4!(2)!
4x3x2x1x2x1
ORDERED ARRANGEMENTS
Sometimes we need to consider how many different ways n items can be ordered.
For example, how many different ways can 8 people be seated around a table? For
the first seat there are 8 choices, for the second seat there are 7 choices, for the
third seat there are 6 choices, etc.
The number of possible ordered arrangements is :
8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 or 8! ( “ 8 factorial)
.
Ordered arrangements are
Permutations
PERMUTATIONS
How many different ways can 10 sprinters finish an Olympic trial? For 1 st place there
are 10 choices, for 2nd place there are 9 choices, for 3rd place there are 8
choices, for 4th place there are 7 choices, for 5 th place there are 6 choices, for 6th
place there are 5 choices, for 7th place there are 4 choices, for 8th place there are
3 choices, for 9th place there are 2 choices, and for the 10 th and final place there
is only one remaining choice.
The number of possible ordered arrangements is
10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 of 10! (“10 factorial”)
CONSIDER ANOTHER SAMPLE SPACE
How many ways can three members of the track team finish a race in the top three
positions?
A simple illustration shows there are just 6 possible combinations.
COUNTING RULE FOR PERMUTATIONS
But what happens when only some of the objects will be chosen?
The number of ways to arrange in order n distinct objects , taking them r at a time, is
n Pr
or Pn,r = n!
(n-r)!
Where n and r are whole numbers.
n≥r
and n! (“n factorial”) = n x (n – 1) x (n – 2) x …. 1
LET’S TAKE ANOTHER LOOK AT THE SAMPLE
SPACE
How many ways can three members of the track team finish a race in the top three
positions?
Using the permutation formula nPr or Pn,r we calculate:
The number of permutations of 3 distinct players , taken 3 at a time, is
3!
= 3!___ = 3 x 2 x 1 = 6
3P3 or P3,3 =
(3 - 3)!
0!
1
CONSIDER ANOTHER SAMPLE SPACE
How many ways can these five members of the track team finish a race in the top
three positions?
Using the permutation formula nPr or Pn,r we calculate:
The number of permutations of 5 distinct players , taken 3 at a time, is
5!
= 5!___ = 5 x 4 x 3 x 2 x 1 = 60
5P3 or P5,3 =
(5 - 3)!
2!
2x1
AND ANOTHER SAMPLE SPACE
How many ways can these six members of the track team finish a race in the top 3
positions?
Using the permutation formula nPr or Pn,r in our track team example,
we calculate:
The number of permutations of 6 distinct players , taken 3 at a time, is
6!
= 6!___ = 6 x 5 x 4 x 3 x 2 x 1 = 120
6P3 or P6,3 =
(6 - 3)!
3!
3x2x1
COMBINATIONS AND PERMUTATIONS
st
1
nd
2
3rd
No Order
n Cr
or Cn,r =
n!
r!(n-r)!
nPr
or Pn,r = n!
(n-r)!
CREDITS
Designer :
Cynthia Toliver
Co- Stars :
Combinations, nCr
Permutations, nPr
Supporting Characters
Boy
Bro
Wry
Con
DJ
Dude