Chapter 7-1 (Day 1) Permutations and Combinations
Obj: To solve problems involving the Fundamental Counting Principal, and involving Combinations and
Permutations
Why? - Permutations are used to determine the number of ways to select and arrange artwork to give a new look each day (ex 2B)
In the warm up you used a tree diagram to find/list the number of combinations of a group of objects.
We can also use the Fundamental Counting Principal.
Fundamental Counting Principal – If event 𝑀1 can occur in 𝑚1 ways, and is followed by an
independent event 𝑀2 that can occur in 𝑚2 ways (and so on…), then the event 𝑀1
followed by 𝑀2 (and so on…) can occur in 𝑚1 ∙ 𝑚2 ∙ … ∙ 𝑚𝑛 ways.
Ex. 1 Using the Fundamental Counting Principle
A – For the lunch special, a customer can choose an entrée (Pizza, Chicken, or Beef), a drink
(Milk, Juice, Punch, or Ice Tea) and one side dish (Coleslaw, Salad, or Fruit). How many meal
choices are there?
________ x ________ x ________ =
B – In Utah, a license plate consists of 3 digits followed by 3 letters. The letters I, O, and Q are not used,
and each digit or letter may be used more than once. How many different license plates are possible?
C – During the first week of summer, the Allwood Movie Theater Complex has a tough decision to make.
There will be 8 new blockbuster films coming out, but they only have a 5 screen theater. Since the
screens are in rooms with different seating capacity, they want to make the best decision of what films
and which rooms to show them in to make the most amount of income from that first week. How many
different arrangements are there for the 5 screens to show the 8 movies?
____
1st
____
2nd
____
3rd
____
4th
____
5th
Hint: after one movie is chosen for the first
screen, there are only 7 choices left, until the
5th screen
The 5 movie screens above are considered in a certain order. When a group or objects or
people are arranged in a certain order, the arrangement is called a permutation.
A Permutation – is a selection of a group of objects in which order is important.
𝒏 𝑭𝒂𝒄𝒕𝒐𝒓𝒊𝒂𝒍 𝒐𝒓 𝒏! – The factorial of a number is the product of the natural numbers
less than or equal to the number. 𝑛! = 𝑛 ∙ (𝑛 − 1) ∙ (𝑛 − 2) ∙ (𝑛 − 3) ∙ … ∙ 1)
ex: 6! = 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 720
Notice from the movie theater example, that 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 is the product of the first 5
factors of 8! If we think of it in a different way….
8∙7∙6∙5∙4∙
3∙2∙1
8!
𝑜𝑟
3∙2∙1
3!
The number of ways to arrange 8 things taken 5 at a time is written as 𝑃(8, 5) or 8P5.
Thus 𝑃(𝑛, 𝑟) or nPr is read “the permutation of n objects taken r at a time”.
The number of permutations of n objects taken r at a time is defined as follows:
𝑷(𝒏, 𝒓)
or
nPr =
(
𝒏!
𝒏−𝒓)!
So again from above…the number of permutations of 8 items taken 5 at a time is
8P5
=(
8!
8−5)!
=
8!
3!
=
Ex 2. Finding Permutations:
A - How many ways can a club select a president, a vice president, and a secretary from a group of 5
people?
B – An art gallery has 9 fine-art photographs from an artist and will display 4 from the left to right along
a wall. In how many ways can the gallery select and display the 4 photographs?
C – How many ways can a stylist arrange 6 of 11 total vases from left to right in a store display?
D – A group of 5 teens went to the movie theater. They found a row with 7 empty seats. How many
different ways can the teens be seated in the row?
7.1 Homework– (day 1) pg 486, 2-6, 9- 13
Chapter 7-1 (Day 1) Warm - up
Make a tree diagram to illustrate the following:
At Ralph’s, they have a special where you can order a large pizza with your
choice of 2 kinds of crust (thin or crispy), 2 kinds of style (traditional or white)
and your choice of one meat topping (meatball, sausage, or pepperoni).