NAME ____________________________________________ DATE _____________________________ PERIOD _____________
12-6 Study Guide and Intervention
Permutations and Combinations
Permutations An arrangement or listing in which order or placement is important is called a permutation. For example,
the arrangement AB of choices A and B is different from the arrangement BA of these same two choices.
The number of permutations of n objects taken r at a time is
!!
P(n, r) =
.
Permutations
(! ! !)!
Example 1: Find P(6, 2).
P(n, r) =
P(6, 2) =
=
=
!!
(! ! !)!
!!
(! ! !)!
!!
!!
! · ! · ! · ! · ! · !
! · ! · ! · !
= 6 · 5 or 30
Permutation Formula
n = 6, r = 2
Simplify.
Definition of factorial
Simplify.
There are 30 permutations of 6 objects taken 2 at a time.
Example 2: PASSWORDS A specific program requires the user to enter a 5-digit password. The digits cannot
repeat and can be any five of the digits 1, 2, 3, 4, 7, 8, and 9.
a. How many different passwords are possible?
b. What is the probability that the first two digits are
odd numbers with the other digits any of the
remaining numbers?
P(n, r) = n!(n – r)!
P(7, 5) =
=
!!
P(first two digits odd) =
(! ! !)!
! · ! · ! · ! · ! · ! · !
! · !
= 7 · 6 · 5 · 4 · 3 or 2520
!"#$%& !" !"#$%"&'( !"#$!%&'
!"#$%& !" !"##$%&' !"#$!%&'
favorable
outcomes:
There are 4 choices
for the first 2 digits
and 5 choices for the
remaining 3 digits.
P(4, 2) · P(5, 3)
possible
outcomes:
There are 7 choices
for the 5 digits.
P(7, 5)
There are 2520 ways to create a password.
The probability is
!(!,!) · !(!,!)
!(!,!)
=
!"#
!"!#
or about 28.6%.
Exercises
Evaluate each expression.
1. P(7, 4)
2. P(12, 7)
3. P(9, 9)
4. CLUBS A club with ten members wants to choose a president, vice-president, secretary, and treasurer. Six of the
members are women, and four are men.
a. How many different sets of officers are possible?
b. What is the probability that all officers will be women.
Chapter 12
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Glencoe Algebra 1
NAME ____________________________________________ DATE _____________________________ PERIOD _____________
12-6 Study Guide and Intervention (continued)
Permutations and Combinations
Combinations An arrangement or listing in which order is not important is called a combination.
For example, AB and BA are the same combination of A and B.
The number of combinations of n objects taken r at a time is
!!
C(n, r) =
.
Combinations
! ! ! ! !!
Example: A club with ten members wants to choose a committee of four members. Six of the members are women,
and four are men.
a. How many different committees are possible?
C(n, r) =
C(10, 4) =
=
!!
Combination Formula
! ! ! ! !!
!"!
n = 10, r = 4
!" ! ! ! !!
!" · ! · ! · !
Divide by the GCF 6!.
!!
= 210
Simplify.
There are 210 ways to choose a committee of four when order is not important.
b. If the committee is chosen randomly, what is the probability that two members of the committee are men?
Probability (2 men and 2 women) =
!"#$%& !" !"#$%"&'( !"#$!%&'
!"#$%& !" !"##$%&' !"#$!%&'
favorable outcomes: There are 4 choices for the 2 men
C(4, 2) · C(6, 2)
and 6 choices for the 2 remaining spots.
The probability is
!(!,!) · !(!,!)
!(!",!)
=
!!
!"#
or about 42.9%.
Exercises
Evaluate each expression.
1. C(7, 3)
2. C(12, 8)
3. C(9, 9)
4. COMMITTEES In how many ways can a club with 9 members choose a two-member sub-committee?
5. BOOK CLUBS A book club offers its members a book each month for a year from a selection of 24 books.
Ten of the books are biographies and 14 of the books are fiction.
a. How many ways could the members select 12 books?
b. What is the probability that 5 biographies and 7 fiction books will be chosen?
Chapter 12
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Glencoe Algebra 1