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# permutations_and_combinations

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• pg 1
```									         Activating Activity
• If ice cream sundaes come in 5 flavors
with 4 possible toppings, how many
different sundaes can be made with one
flavor of ice cream and one topping?

1
Counting Principle
Permutations
Combinations

How do you use the counting
principle, permutations and
combinations to predict outcomes
of different events or
occurrences?
2
The Fundamental Counting
Principle:

If there are a ways for one
activity to occur, and b ways for a
second activity to occur, then
there are a • b ways for both to
occur.

3
Examples:
1. Activities: roll a die and flip a coin
There are 6 ways to roll a die and 2 ways to
flip a coin.
There are 6 • 2 = 12 ways to roll a die and
flip a coin.

2. Activities: draw two cards from a standard
deck of 52 cards without replacing the cards
There are 52 ways to draw the first card.
There are 51 ways to draw the second
card.
There are 52 • 51 = 2,652 ways to draw the
two cards.                     4
The Counting Principle also works for more than
two activities.
3. Activities: a coin is tossed five times
There are 2 ways to flip each coin.
There are 2 • 2 • 2 • 2 •2 = 32 arrangements of

4. Activities: a die is rolled four times
There are 6 ways to roll each die.
There are 6 • 6 • 6 • 6 = 1,296 possible outcomes.

Remember: The Counting Principle is easy! Simply
MULTIPLY the number of ways each activity can occur.   5
A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3
choices of toppings (no butter, butter, extra butter). How many possible
ways can a bag of popcorn be purchased?

Colors: white, beige, pink, yellow, blue
Sizes: 4, 5, 6, 7, 8
Extras: tassels, striped laces, bells
Assuming that all skates are sold with ONE extra, how many
possible arrangements exist?

6
numbers. There are 26 letters and the letters may be
repeated. There are 10 digits and the digits may be
repeated. How many possible license plates can be
issued with two letters followed by three numbers?

The ice cream shop offers 31 flavors. You order a double-scoop cone. In
how many different ways can the clerk put the ice cream on the cone if you
wanted two different flavors?

7
The local Family Restaurant has a daily breakfast special in
which the customer may choose one item from each of the
following groups:
Breakfast   Accompan
Juice
Sandwich     iments
egg and     breakfast
orange
ham        potatoes
cranberry
egg and    apple slices
tomato
bacon      fresh fruit
apple
egg and        cup
grape
cheese        pastry

a.) How many different breakfast specials
are possible?
b.) How many different breakfast specials
without meat are possible?

8
Permutation:
A set of objects in which position (or order) is important.
To a permutation, the trio of Brittany, Alan and Greg is
DIFFERENT from Greg, Brittany and Alan. Permutations are
persnickety (picky).

Combination:
A set of objects in which position (or order) is NOT
important.
To a combination, the trio of Brittany, Alan and Greg is THE
SAME AS Greg, Brittany and Alan.

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Let's look at which is which:

Permutation        versus   Combination

1. Picking a team captain,           1. Picking three team members
pitcher, and shortstop from a        from a group.
group.

2. Picking your favorite two         2. Picking two colors from a
colors, in order, from a color       color brochure.
brochure.

3. Picking first, second and         3. Picking three winners.
third place winners.

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Formulas:

A permutation is the
choice of r things from
a set of n things
without replacement
and where the order
matters.

Remember: n is the total number of objects
r is the number of objects chosen (want)

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1.   Compute:   5   P5

2.   Compute:   6P2

3. Find the number of ways to arrange 5 objects that are chosen from a set of 7 different
objects.

4. What is the total number of possible 5-letter arrangements of the letters w, h, i, t, e, if each
letter is used only once in each arrangement?

5. How many different 3-digit numerals can be made from the digits 4, 5, 6, 7, 8 if a digit can
appear just once in a numeral?

12
1.   Compute:     5   P5     5 · 4 · 3 · 2 · 1 = 120

2.   Compute:     6P2         6 · 5 = 30                   or
multiply by two factors
of the factorial, starting with 6

3. Find the number of ways to arrange 5 objects that are chosen from a set of 7 different
objects.

7P5    = 7·6·5·4·3 = 2520          or

4. What is the total number of possible 5-letter arrangements of the letters w, h, i, t, e, if each
letter is used only once in each arrangement?

5 P5    = 5·4·3·2·1 = 120            or                             or   simply 5!

5. How many different 3-digit numerals can be made from the digits 4, 5, 6, 7, 8 if a digit can
appear just once in a numeral?

5 P3    = 5·4·3 = 60             or

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A permutation is an arrangement of
objects in specific order.
The order of the arrangement is
important!!

Consider, four students walking toward their
school entrance. How many different ways could
they arrange themselves in this side-by-side
pattern?
1,2,3,4   2,1,3,4   3,2,1,4   4,2,3,1
1,2,4,3   2,1,4,3   3,2,4,1   4,2,1,3
1,3,2,4   2,3,1,4   3,1,2,4   4,3,2,1
1,3,4,2   2,3,4,1   3,1,4,2   4,3,1,2
1,4,2,3   2,4,1,3   3,4,2,1   4,1,2,3
1,4,3,2   2,4,3,1   3,4,1,2   4,1,3,2

The number of different arrangements is 24 or 4! = 4 •
3 • 2 • 1. There are 24 different arrangements, or
permutations, of the four students walking side-by-side.
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In how many ways can 3 different vases be arranged on a tray?

15
A combination is the choice of r things from a set of n
things without replacement and where order does not
matter.

16
Example 1:
Evaluate     :

Notice how the cancellation occurs, leaving only
2 of the factorial terms
in the numerator. A pattern is emerging ... when
finding a combination
such as the one seen in this problem, the
second value (2) will tell you
how many of the factorial terms to use in the
numerator, and the
denominator will simply be the factorial of the
second value (2).

17
Heather has finally narrowed her clothing choices for the big
party down to 3 skirts, 2 tops and 4 pair of shoes. How many
different outfits could she form from these choices?

18
A coach must choose five starters from a team of 12
players. How many different ways can the coach choose the
starters?

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Joleen is on a shopping spree. She buys six tops, three shorts and 4 pairs of sandals. How many
different outfits consisting of a top, shorts and sandals can she create from her new purchases?
(6)(3)(4) = 72 possible outfits

What is the total number of possible 4-letter arrangements of the letters
m, a, t, h, if each letter is used only once in each arrangement?
or                                     or simply 4!

20
3-2-1 Ticket out the door
• Name three different terms used to predict
outcomes of events.

• Name two formulas used today.

• Name one way to tell which concept you
will choose to use to predict outcomes of
events.
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