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12-6 Study Guide and Intervention
Permutations and Combinations
Permutations An arrangement or listing in which order or placement is important iscalled a permutation. For example, the arrangement AB of choices A and B is differentfrom the arrangement BA of these same two choices.
Permutations / The number of permutations of n objects taken r at a time isP(n, r) = .
Example 1: Find P(6, 2).
P(n, r) = Permutation Formula
P(6, 2) = n = 6, r = 2
= Simplify.
= Definition of factorial
= 6 · 5 or 30 Simplify.
There are 30 permutations of 6 objects taken 2 at a time.
Example 2: PASSWORDS A specific program requires the user to enter a5-digit password. The digits cannot repeat and can be any five of the digits1, 2, 3, 4, 7, 8, and 9.
Chapter 1235Glencoe Algebra 1
NAME ______DATE______PERIOD ______
a. How many differentpasswords are possible?
P(n, r) = n!(n –r)!
P(7, 5) =
=
= 7 · 6 · 5 · 4 · 3 or 2520
There are 2520 ways to createa password.
b. What is the probability that the first two digitsare odd numbers with the other digits any ofthe remaining numbers?
P(first two digits odd) =
favorableThere are 4 choicesP(4, 2) · P(5, 3)
outcomes:for the first 2 digits
and 5 choices for the
remaining 3 digits.
possibleThere are 7 choices P(7, 5)
outcomes:for the 5 digits.
The probability is = or about28.6%.
Chapter 1235Glencoe Algebra 1
NAME ______DATE______PERIOD ______
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.
12-6 Study Guide and Intervention(continued)
Permutations and Combinations
Combinations An arrangement or listing in which order is not important is called acombination.
For example, AB and BA are the same combination of A and B.
C(n, r) = .
Example: A club with ten members wants to choose a committee of fourmembers. Six of the members are women, and four are men.
a. How many different committees are possible?
C(n, r) = Combination Formula
C(10, 4) = n = 10, r = 4
= Divide by the GCF 6!.
= 210Simplify.
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 membersof the committee are men?
Probability (2 men and 2 women) =
favorable outcomes:There are 4 choices for the 2 menC(4, 2) · C(6, 2)
and 6 choices for the 2 remainingspots.
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-membersub-committee?
5. BOOK CLUBS A book club offers its members a book each month for a year from aselection 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 1236Glencoe Algebra 1