Permutations & Combinations Packet

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Intro to Probability; Permutation & Combinations
Class Goals/Summary:
● Understanding the concept of probability
● Permutations and Combinations
________________________________________________________________________________________
As it turns out, pretty much everything in the world–every decision–is based on some form of probability.
Concept: Loosely, probability is ___________________________________________________________.
As easy as that definition is to write down and say, theoretically probability is a little bit tricky.
For example, everyone knows that if we flip a coin, the probability of getting heads should be 50%. But what if
you flip two coins–what are the probability of getting heads the second flip? What is the probability of getting
heads the second flip if you already got heads the first flip? What if your coin is weighted? What about the
probability of the coin landing on its edge?
As you can tell, lots of factors can contribute to and alter the probability of an event happening (or not
happening). As pupils of math though, we try to find as many ways to relate concrete mathematics to the
abstractness of probability (or “guessing”, as I like to call it). To get to calculating probabilities, we first need to
understand some key concepts:
Key Concepts/Definitions:
● Event:
● Permutation:
● Combination:
To find the probability of an event, permutations and combinations are used often. The most important is the
underlying counting processes.
They are confusing to get mixed up on which one is which! Rule of thumb:
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__________________________________________________________________________________________
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Permutations: The Hairy Details
Think about permutations as all possible ways to do somethings. Let us say we have 8 people:
Ally
Barbara
Casey
Daniel
Evan
Frank
George
Hector
All eight people are good friends. They will all enter the 2018 Winter Olympics.
Question: How many ways can we award a 1st, 2nd, and 3rd place prize among eight contestants. (Gold /
Silver / Bronze)
Order matters in this question because the order we hand out these medals matter. Let us breakdown the events:
● Gold medal: We have 8 choices – A B C D E F G H (I made the names match up with letters!). Let say
A wins Gold.
● Silver medal: If A wins Gold, then we have 7 choices left – B C D E F G H. Let us say B wins Silver.
● Bronze medal: If B wins Silver, then we have 6 choices left- C D E F G H. Let us say C wins Bronze.
We picked certain amount of people to win. We had 8 choices total, then 7, then 6. So 8 𝑥 7 𝑥 6 = 336
possible ways we can award a 1st, 2nd, and 3rd place prize among eight contestants.
Let us look at some hairier details. Again, we had to order 3 people out of 8. We began with all 8 people. Then,
we took away one person at a time (7, then 6) until we ran out of medal. This product of positive integers is
called factorial.
8! = 8 𝑥 7 𝑥 6 𝑥 5 𝑥 4 𝑥 3 𝑥 2 𝑥 1 = 40, 320 total ways
But that is way too much! We only want 8 𝑥 7 𝑥 6 since there are three prizes.
Question: How can we “stop” the factorial at 5?
This is where permutation gets awesome: Notice how we got rid of 5 𝑥 4 𝑥 3 𝑥 2 𝑥 1 which is 5 factorial! A
factorial is the product of an integer and all the integers below it.
So, if we do
8! 8 𝑥 7 𝑥 6 𝑥 5 𝑥 4 𝑥 3 𝑥 2 𝑥 1
=
= 8𝑥7𝑥6
5!
5𝑥 4 𝑥 3 𝑥 2 𝑥 1
2
Why did we use the number 5? Because after we picked 3 medals, we have 5 people left over from 8. A better
way to write this would be
The above expression is just a fancy way of saying “Use the first 3 numbers of 5”. Consider a more general
scenario where if we have n items total and want to pick k in a certain order, we get
The above expression is the fancy permutation expression!
Exercises:
Find the number of possible ways to do these scenarios.
•
Picking a President, Vice President, and Secretary from a
group of 10.
•
Listing your favorite 3 desserts, in order, from a menu of 5
items.
•
Picking your top 5 favorite ice cream flavors, in order,
from an ice cream menu of 15.
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Combinations
Combinations are a lot easier going because order does not matter. You can mix things up and they all represent
the same things. Let us say we cannot afford Gold, Silver, and Bronze medals. We can only afford empty tin
cans.
Question: How many ways can I give 3 tin cans to 8 people?
In this case, the order we pick people does not matter. If we give a tin can to Ally, Barbara, and Casey, it is the
same as giving a tin can to Casey, Ally, and Barbara. So, Ally, Barbara, and Casey = Casey, Ally, and Barbara.
Notice we have some repetitions here.
Let us figure out how many ways we can rearrange 3 people. We have 3 choices for the first person, 2 choices
for the second person, and 1 choice for the last person. So, we have 3 𝑥 2 𝑥 1 ways to rearrange 3 people.
This sounds like permutations!! Yes, it is! If we have N people, and you want to know how many arrangements
there are for all of them, it is just N!
So, if we have 3 tin cans to give away, there are 3! Or 6 total ways for every choice we pick. If you want to
understand how many combinations we have, we can create all the permutations and divide by all the
repetitions.
From the above permutation example, we have 336 permutations we can award a 1st, 2nd, and 3rd place prize
among eight contestants. We can divide by the 6 repetitions for each permutation.
336
= 56 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠
6
Notice we have the permutation on the numerator and divided by the 6 repetitions. This helps us derive the
combinations formula.
We have
𝑛!
(𝑛 − 𝑘)!
𝑘!
=
We can divide the fractions.
𝑛!
(𝑛−𝑘)!𝑘!
This gets us our Combination expression.
The formula means “find all ways to pick k people from n and divide by the k”.
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Exercises: Find the number of possible ways to do these
scenarios.
•
Picking a basketball team of 3 people from a group of
10.
•
Choosing 3 desserts from a menu of 5.
•
Choosing a baseball team of 10 people from a group of
30.
Combinations and permutations are a matter of picking an object and arranging those objects depending on if
order matters or not.
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