# Counting_ Permutations_ and Combinations

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```					                  Activator
   Ice cream activity
Counting, Permutations,
and Combinations
Unit 4 Part 1
Warm Up
 How many different ways can you arrange
the letters in REBEL?
 How many different ways can you arrange
the letters in BERRIEN?
Basic (or Fundamental)
Counting Principle
 States that if one event can occur in m
ways, and a second event can occur in n
ways, then the number of ways that both
events can occur is m times n ways.
 Example: Warm Up Examples (class
discussion)
Basic (or Fundamental)
Counting Principle
 And…if there is a third event that can
occur in p ways, the number of ways that
all three events can occur is m times n
times p ways.
 This principle can be extended for any
number of events.
Graphic Organizer #1
More Basic (or Fundamental)
Counting Principle examples…
   You are asked to create a password for
your email account. The password must
start with 1 digit (0-9) and be followed by
three letters (A-Z). How many different
possible passwords are there?

number      letter     letter    letter
Solution
 The digit at the beginning has 10 different
possibilities (0-9)
 The second position can be any letter A-
Z…26 possibilities
 The third position can be any letter A-
Z….26 possibilities
 The fourth position can be any letter A-
Z…26 possibilities
 10 * 26 * 26 * 26 = 175,560
Example 2
 Now, let’s try that same password…except
none of the letters can be repeated!
 10 possibilities
 26 possibilities
 25 possibilities
 24 possibilities
 10 * 26 * 25 * 24 = 156,000
Tell the number of possibilities
 1. You can go to any of three beaches in
2 choices to make:
Choice 1: 3 choices
Florida and choose scuba diving, 3 choices
Choice 2:
of
parasailing, or surfing at the beach 9 your
3•3=
choice. How many different vacation
options do you have?
 2. You have 10 shirts, 8 pairs of pants,
3 Choices to make:
and 5 pairs of shoes. How many different
Choice 1: 10 shirts
Choice 2: 8 pants
outfit combinations could there be?
Choice 3: 5 shoes

10•8•5 = 400
 How many different ways can3•2•1 = 6
you
rearrange the word CAT?
 Try HORSE        5•4•3•2•1 = 120

   Try HIPPOPOTAMUS
12•11•10•9•8•7•6•5•4•3•2•
1=
479,001,600
   Try your name!
   Complete Counting Principle Practice WS
(1-5)
Factorials
 Factorials are indicated by the “!” symbol.
 Whenever you see a ! with a number that
means you multiply the number with every
number below it.
 Ex. 5! Means 5 * 4 * 3 * 2 * 1 = 120
 7! Means 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
 3! Means 3 * 2 * 1 = 6
 0! Always equals 1
Operations with Factorials
Factorial practice WS
   Example: 1! + (6-2)!

   Example: (7-3)! ÷ 2!

   Example: 6! ÷ 3!
Practice
   Complete (3-10)
What would you do here?
   How many ways can you choose 3
pictures out of a group of 5 to arrange in
a row on your wall?

Because we are
choosing a group of 3
from a big group of
5…we are not
arranging each of the
objects…This will take
a special formula…
Important Questions
 We are about to learn some formulas. To
use these formulas, we need to answer
some questions first.
 How many things are you choosing from?
This answer will be your “n” in the
formulas.
 How many things can you choose? This
will be your “r” in the formulas.
Identify n and r
   Thirty five people try out for basketball,
and only 15 can be on the team.
n = 35
r = 15

   You can choose three toppings for your
pizza. The pizza restaurant now offers 20
different toppings.
n = 20
r=3
Does order matter?
 What if 25 people are running a race, and we
want to know how many possibilities there are
for the top three finishers. Does order matter?
 Of course it does!

 If you get three vegetables on your plate and
you get to choose from ten, does order matter?
 No…you still are going to get each of those
three vegetables.
Permutations and Combinations
 Permutations occur when order does
matter…like the first three finishers of a
race or setting your locker combination or
password
 Combinations occur when order does not
matter…like what three vegetables to
choose
Permutations
 Use when the order of the items is
important, as in a race or arranging things
 The formula for a permutation is:

nPr = __ n!__
(n-r)!
Complete graphic organizer #3
on Permutations
Practice Using Permutation
Formula WS
Permutation Practice
 How many different ways can a
chairperson and an assistant chairperson
be selected for a research project if there
are seven scientists available?
 What is your “n” value?
 What is your “r” value?
 Use the permutation formula to find your
answer.
More practice…
 How many different ways can four books
be arranged on a shelf if they are selected
from eight books?
 What is your “n” value?
 What is your “r” value?
 Use the formula and find the total number
of combinations.
Combinations
 Use when items represents different
scenarios, such as selecting two types of
yogurt from a dairy case from a selection
of nine
 The formula for a combination is:

nCr = __ n!__
r!(n-r)!
Graphic Organizer #4
Combination example problems
 You are to select two types of yogurt from
the dairy case from a selection of 9.
 What is your “n” value?
 What is your “r” value?
 Use the combination formula to find the
number of combinations possible.
Another example problem…
 Select three members from your class to
work specific homework problems on the
board.
 What is your “n” value?
 What is your “r” value?
 Use the combination formula to find the
total number of combinations possible.
Complete Combination
Practice WS

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