Algebra 2
19.2 Notes
19.2 Permutations
Date: __________
Learning Target C: I can find the number of permutations of a given number of objects.
A permutation is a selection of objects from a group in which order is important. For example, there
are 6 permutations of the letters A, B and C.
You can find the number of permutations with the Fundamental Counting Principle.
Example 1. Use the Fundamental Counting Principle to find the number of permutations for each
situation.
A) Jerry is going on vacation to Mexico and decided to pack light, so he didn’t need to check a bag. He
is bringing 2 pairs of shoes, 2 pairs of shorts, 1 pair of pants, and 4 shirts. How many different outfits
can he make given the items he has packed?
B) An ice-cream shop lets you build your own sundae with the options shown in the menu below. How
many different sundaes are possible?
Container
Paper Dish
Waffle Dish
Ice-Cream Flavor
Vanilla
Chocolate
Strawberry
White Chocolate
Butterscotch
Topping #1
Chocolate Sauce
Hot Fudge
Caramel
Butterscotch
Strawberry Sauce
Topping #2
Rainbow Sprinkles
Spanish Peanuts
Crushed Peanuts
C)The 7 members of the Chess Club are lining up for their yearbook picture. In how many different
orders can they line up?
D) The club is holding elections for a president, a vice president and a treasurer. How many different
ways can these positions be filled?
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Algebra 2
19.2 Notes
The results of the previous example can be generalized to give a formula for permutations. To
do so, it is helpful to use factorials. For a positive integer n, n factorial, written _____, is defined as
follows:
𝑛! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) × … × 3 × 2 × 1
That is, n! is the product of n and all the positive integers less than n. Note that 0! is defined to be 1.
In example D on the previous page the number of permutations of the 7 objects taken 3 at a time was:
7 ×6 ×5=
7 ×6 ×5 ×4 ×3 ×2 ×1
7!
7!
=
=
(7 − 3)!
4 ×3 ×2×1
4!
Use the formula for permutations to find each.
A) A research laboratory requires a four-digit security code to gain access to the facility. A security
code can contain any of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, but no digit is repeated. How many
different security codes are possible?
B) How many 4-digit security codes are possible if 0 cannot be used in the code?
C) A certain motorcycle license plate consists of 5 digits (0-9) that are randomly selected. No digit is
repeated. How many license plate numbers are possible?
D) There are 8 finalists in the 100-meter dash at the Olympic Games.
a. How many different orders can the finalists finish the race?
b. Suppose 3 of the finalists are from the United States, how many different ways can the United
States win the top three places?
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Algebra 2
19.2 Notes
Finding the Number of Permutations with Repetition
Learning Target D: I can find permutations with repetition.
Up to this point, the problems have focused on finding the permutations of distinct objects. If some of
the objects are repeated, this will reduce the number of permutations that are distinguishable.
For example, here are the permutations of the letters A, B and C.
Next, here are the permutations of the letters M, O, and M. Bold type is used to show different
positions of the repeated letter.
Shown without bold type….
Notice there are only ______ distinguishable permutations. This can be generalized with a formula for
permutations with repetition.
Find the number of permutations.
A) How many different permutations are there of the letters in the word ARKANSAS?
B) One of the zip codes for Anchorage, Alaska, is 99522. How many permutations are there of the
numbers in this zip code?
C) How many different permutations can be formed using all the letters in MISSISSIPPI?
D) One of the standard telephone numbers for directory assistance is 555-1212. How many different
permutations of this telephone number are possible?
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