Permutations
Date: ______________
I can…use counting techniques to find all of the possible ways to complete different tasks or choose items from a list.
Key words: fundamental counting principle, permutations, factorial
How does the
Try this: Suppose you are choosing an outfit. You can select blue or black pants. You have a choice
Fundamental
of red, green, or yellow shirt. How many total outfits can you make? Draw a diagram to answer this
Counting
question.
Principle help
to find total
combinations?
The Fundamental Counting Principle says that if event M occurs in m ways and event N occurs in n
ways, then event M followed by event N can occur m · n ways.
Example: 4 entrees and 6 desserts = 4 · 6 = 24 meals
Example:
Example:
1. International calls require the use of a country code. Many country codes are 3-digit numbers. Country codes do
not begin with 0 or 1. There are no restrictions on the second and third digits. How many different 3-digit country
codes are possible?
2. To make an entry code, you first need to choose a single-digit number and then two letters, which can repeat. How
many entry codes can you make?
How do
permutations
help to find
total
combinations?
Try this: Emily, Hannah, Maci, and Kiley all need to get in a line. How many different ways can they
get in line? Draw a diagram to answer this question.
A permutation is an arrangement of items in which the order of the objects is important.
Example:
Example:
You can use factorial notation to find the number of permutations.
Example: 5! is read as “five factorial” and is equal to 5 · 4 · 3 · 2 · 1
Example: 7! =
**Zero factorial (0!) is equal to 1.
3. In how many ways can you arrange 10 books on a shelf?
4. 6! =
What is
permutation
notation?
5. 9! =
Try this: You have six people - Abby, Taylor, Jordan, Ben, Matthew, and Claire - who may be part of a
three-person committee. From these six people, how many different groups of three can be made?
Draw a diagram to answer this question.
The number of permutations of n items of a set arranged r items at a time is
nPr
𝑛!
= (𝑛 − 𝑟)! for 0 ≤ r ≤ n
Example:
Example:
6. Twelve swimmers compete in a race. In how many possible ways can the swimmers finish first, second, and third?
7. 10P6
8. 11P5
Homework: Permutations
Name: _____________________________________
1. Suppose that a computer generates passwords that begin with a letter followed by two digits, like L17. The same
digit can be used more than once. How many different passwords can the computer generate?
2. If you roll a die and flip a coin, how many different combinations are possible?
3. You go to Sweet Frog for dessert. There are 9 different flavors of ice cream. There are 23 topping choices. If you
choose only one from each category, how many different desserts are possible?
4. In one game, a code made using different colors is created by one player (codemaker), and the other player (the
codebreaker) tries to guess the code. The codemaker gives hints about whether the colors are correct and in the right
position. The possible colors are blue, yellow, white, red, orange, and green. How may 4-color codes can be made if
the colors cannot be repeated?
4. There are seven friends, but only three chairs. How many possible ways can these people sit in the three chairs?
5. Find the value: 8! =
6. Find the value: 9P4 =
Homework: Permutations
Name: _____________________________________
1. Suppose that a computer generates passwords that begin with a letter followed by two digits, like L17. The same
digit can be used more than once. How many different passwords can the computer generate?
2. If you roll a die and flip a coin, how many different combinations are possible?
3. You go to Sweet Frog for dessert. There are 9 different flavors of ice cream. There are 23 topping choices. If you
choose only one from each category, how many different desserts are possible?
4. In one game, a code made using different colors is created by one player (codemaker), and the other player (the
codebreaker) tries to guess the code. The codemaker gives hints about whether the colors are correct and in the right
position. The possible colors are blue, yellow, white, red, orange, and green. How may 4-color codes can be made if
the colors cannot be repeated?
4. There are seven friends, but only three chairs. How many possible ways can these people sit in the three chairs?
5. Find the value: 8! =
6. Find the value: 9P4 =