11.6: Counting Principles, Combinations, and Permutations
Consider two scenarios:
i.
When Mr. Cawelti goes out to the club, he chooses from 3
shirts, 2 pairs of pants, and 4 pairs of shoes (let’s ignore those
occasions when he chooses to wear his Fozzie Bear costume).
ii.
After turning 16, Soha is considering all of her luxury car
options for her new “whip”. She must decide on a type (sedan,
SUV, or truck), a color (green, blue, red, white, or black), and a
transmission type (manual or automatic).
In each case, how many different options exist?
1.
Jaden’s WIFI password is 10 characters in length, all of which are
lower case letters. How many different possibilities exist?
2.
In Colorado, license plates consist of 6 characters: 3
digits followed by 3 letters. After having his house
toilet papered, Mr. Cawelti attempted to identify the
license plate of the perpetrators as they sped away. He saw the
first digit was a 3, the last digit was not a 0, and the first letter was
a vowel. How many license plate possibilities exist?
Permutations
How do these next scenarios differ from the previous
two?
i.
Mr. Propri has 6 Frozen figurines that he would
like to arrange on his bookshelf. How many
different ways can he do this?
ii.
FHS administration found a list circulating the hallways entitled
“Top 5 Smelliest Teachers at Fairview”, which listed the #1
through #5 smelliest teachers at FHS. While there was much
speculation regarding the contents of the list (my money’s on
Muzny), how many different ways are there to create such a
list out of the 108 total teachers?
A permutation is an ordered arrangement of items that occurs when
 No item is used more than once (repetition not allowed)
 Order matters
Let’s derive a formula:
3.
Mr. Cawelti would like to appoint advanced algebra 2 students to
represent the rest of their algebra constituents on a committee.
One student will hold the rank of Turdburglar, one will be Vice
Turdburglar, and one will be Secretary of Fecal Thievery. How
many different ways can Mr. Cawelti form his committee from the
31 advanced algebra 2 students?
4.
After a heated discussion, a group of Fairview students decided to
rank the 60 Game of Thrones episodes from best to worst. How
many ways are there to do this?
Combinations
How do these next scenarios differ from the previous two?
i.
Henry’s band, Euclidean Ensemble, has 1000 followers on
Twitter. If Henry holds a competition to give away concert
tickets to 5 random followers, how many groups of winners are
possible?
ii.
At this year’s FHS graduation, 15 students will graduate with
honors. How many different groups of honors graduates are
possible if the graduating class is 492 students?
A combination is an ordered arrangement of items that occurs when
 No item is used more than once (repetition not allowed)
 Order does not matter
We already know a formula:
5.
A vegetable bandit ransacked Mr. Propri’s garden, and stole 47
out of his 51 tomatoes. How many different groups of tomatoes
could have been stolen?
6.
Fairview’s Student Council has 8 freshman, 7 sophomores, 7
juniors, and 9 seniors. They would like to send 3 freshman, 3
sophomores, 4 juniors, and 4 seniors to an event. How many ways
can they do this?
7.
The padlock on your locker consists of the integers 0-39, and a
typical locker combination is a sequence of 3 numbers. If you try
one combination at random, what is the probability you will guess
correctly?