Download Permutations - Greer Middle College

8.2 Probability Rules
Name: ________________________
Lesson 5—Counting & Permutations
Fundamental Counting Principle
How probable is it that two separate events occur?
If event 𝐴 can occur ___________ times and event 𝐵 can occur _________ times, then there are
____________ different ways both events can occur.
Examples
1. If an ice cream store offers nine different flavors and three different sizes, then how
many possible combinations of flavors and sizes are there?
2. The menu of a particular restaurant lists three appetizers, eight entrées, four desserts,
and three drinks. Assuming a meal consists of one appetizer, one entrée, one dessert,
and one drink, how many different meals can be ordered?
3. A particular state license plate contains three letters (A-Z) followed by four digits (0-9).
How many unique license plates can be created?
According to the Fundamental Counting Principle, the number of possible license plates
is equal to the product of the possible ways each character can be chosen.
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Factorials
In general, 𝑛! ("_______________________") means the product of all the natural numbers
from 1 to 𝑛; that is,
𝑛! = _______________________
For instance, "𝑓𝑜𝑢𝑟 𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑎𝑙" is written as "4!" and means
1 × 2 × 3 × 4 = 24
Calculator: 𝑀𝐴𝑇𝐻 → ◄ → 4: !
Examples
1. The starting five players of a basketball team are announced one by one at the
beginning of a game. Calculate the total number of different ways the order of players
can be announced.
2. Calculate the total number of ways eight people can be seated at a table that has eight
seats.
3. There are fourteen juniors and twenty-three seniors in the Service Club. The club is to
send four representatives to the State Conference. How many different ways are there
to select a group of four students to attend the conference?
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Practice—Try the questions on the board. Use the space below to work out the problems
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Permutations
How many ways can you arrange a collection of things?
A permutation is a sequence of object in which the __________________ is a defining factor.
Example:
For instance if you are given {𝐴, 𝐵, 𝐶} and asked to identify unique permutations, choosing
two elements at a time, 𝐴𝐵 and 𝐵𝐴 are considered unique permutations.
Although both contain the same two elements, the order in which the elements appear
distinguishes them. No elements are repeated in a unique permutation.
You are commonly asked to calculate the number of permutations that exist for a set
containing _______________ elements if you choose _____________of them at a time.
In the previous example you choose 𝑟 = _________ of the 𝑛 = _________ letters.
The number of possible permutations is defined as ___________________
The equation for finding a permutation is _____________________________
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Calculator:
Turn to page 397 in your textbooks. Find “Calculator Corner” and follow the TI-84 directions to
calculate 28 𝑃3 . Write the steps in the space below.
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Same Example:
To calculate the total number of ways A, B, and C can be arranged in order, two at a time,
evaluate 3 𝑃2 .
𝑃
3 2
= _____________
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The six possible permutations are _____, ______, _____, ______, _____, and ______.
Permutations (cont.)
How many ways can you arrange a collection of things?
Examples
1. If a salesperson is responsible for nine stores, how many different ways can she
schedule visits with five stores this week?
2. Calculate the number of ways eight swimmers can place first, second, or third in a race.
3. A combination lock has a total of 40 numbers on its face and will unlock given the
proper three-number sequence. How many unique combinations are possible?
4. On the way home, you decide to stop by Chewy’s ice cream parlor for a snack. With 28
flavors to choose from, how many different ways are there to order a cone with three
scoops if the order does not matter?
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Ticket out the Door—Complete the following questions on a separate sheet of lined
paper and turn in to Mrs. Diener. Show your process EVEN your calculator work.
1. Rob has 4 shirts, 3 pairs of pants, and 2 pairs of shoes that all coordinate. How many
outfits can you put together?
2. How many ways can 5 paintings be line up on a wall?
3. A lock contains 3 dials, each with ten digits. How many possible sequences of numbers
exist?
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