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.
Slide 4
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.
Slide 5
Toss Coin Head 123456 Tail 123456 Roll Dice TOSS COIN, ROLL
DICE EXAMPLE H1, H2, H3, H4, H5, H6 T1, T2, T3, T4, T5, T6
Resulting Sample Space
Slide 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.
Slide 7
MULTIPLICATION RULE MORE EXAMPLES Fords 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.
Slide 8
MULTIPLICATION RULE MORE EXAMPLES The Old Orange Caf 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.
Slide 9
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
Slide 10
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?
Slide 11
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 1 st 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 2 3 4 5
6
Slide 12
COUNTING RULE FOR COMBINATIONS The number of combinations of n
distinct objects, taking them r at a time, is n C r or C n,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
Slide 13
COMBINATIONS Using the combination formula n C r or C n,r in
our tennis team example, we calculate: The number of combinations
of 4 distinct players, taken 2 at a time, is 4 C 2 or C 4,2 = 4! =
4!___ = 4 x 3 x 2 x 1 = 6 2!(4-2)! 2!(2)! 2 x 1 x 2 x 1 1 2 3 4 5
6
Slide 14
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 n C r or C n,r we calculate: The
number of combinations of 5 distinct players, taken 2 at a time, is
5 C 2 or C 5,2 = 5! = 5!___ = 5 x 4 x 3 x 2 x 1 = 10 2!(5 - 2)!
2!(3)! 2 x 1 x 3 x 2 x 1
Slide 15
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 n C r or C n,r we calculate: The number of
combinations of 6 distinct runners, taken 4 at a time, is 6 C 4 or
C 6,4 = 6! = 6!___ = 6 x 5 x 4 x 3 x 2 x 1 = 15 4!(6 - 4)! 4!(2)! 4
x 3 x 2 x 1 x 2 x 1
Slide 16
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
Slide 17
PERMUTATIONS How many different ways can 10 sprinters finish an
Olympic trial? For 1 st place there are 10 choices, for 2 nd place
there are 9 choices, for 3 rd place there are 8 choices, for 4 th
place there are 7 choices, for 5 th place there are 6 choices, for
6 th place there are 5 choices, for 7 th place there are 4 choices,
for 8 th place there are 3 choices, for 9 th 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)
Slide 18
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.
Slide 19
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 P r or P
n,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
Slide 20
LETS 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 n P r or P n,r we
calculate: The number of permutations of 3 distinct players, taken
3 at a time, is 3 P 3 or P 3,3 = 3! = 3!___ = 3 x 2 x 1 = 6 (3 -
3)! 0! 1
Slide 21
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 n P r or P n,r we calculate: The
number of permutations of 5 distinct players, taken 3 at a time, is
5 P 3 or P 5,3 = 5! = 5!___ = 5 x 4 x 3 x 2 x 1 = 60 (5 - 3)! 2! 2
x 1
Slide 22
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 n P r or P n,r in our track team example, we
calculate: The number of permutations of 6 distinct players, taken
3 at a time, is 6 P 3 or P 6,3 = 6! = 6!___ = 6 x 5 x 4 x 3 x 2 x 1
= 120 (6 - 3)! 3! 3 x 2 x 1
Slide 23
COMBINATIONS AND PERMUTATIONS
Slide 24
CREDITS Designer : Cynthia Toliver Co- Stars : Combinations, n
C r Permutations, n P r Supporting Characters Boy Bro Wry Con DJ
Dude