9.6 Counting Principles

Fundamental Counting Principle- E₁ and E₂ are two different events. E₁ can occur in m₁ different ways and E₂ can occur in m₂ different ways. The number of different ways that the two events can occur together is m₁ x m₂.

The fundamental counting principle is true for any number of events that occur together.

When counting the number of possible outcomes, it is important to know if you are counting combinations or permutations.

When counting permutations, order matters. This means that choosing option 1, 2, and then 3 is different than choosing option 1, 3, and then 2. Lets look at an example…

If you randomly choose 3 digits (between 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), where each digit can only be used once, how many numbers can you make?

This is asking for the number of outcomes. Because order matters, we have to count the number of permutations.

According to the fundamental counting principle, we have to find the product of the number of possible outcomes for each event (m₁ x m₂ x m₃).

1. The first digit can be any of the numbers listed above so there are 10 different possible outcomes.
2. The second digit can be any number listed above other than the first digit so there are 9 possible outcomes.
3. There are 8 possible outcomes for the third digit.

So to find the total number of permutations when you choose three out of ten options...

Our equation for counting the number of permutations is...

Where 

Now let’s look at how to count combinations. When counting combinations, order does not matter. This means that choosing option 1, 2, and then 3 is the same as choosing 1, 3, and then 2. Here’s an example of counting the number of possible combinations…

When ordering a pizza with toppings, it does not matter what order you choose the toppings in. A cheese, ham, pineapple pizza is the same thing as a pineapple, ham, cheese pizza. Let’s say there are ten toppings to choose from. How many ways can you choose 3?

If order mattered we would find the answer like this...

However, because order doesn't matter, this number is too big because it counts

cheese, ham, pineapple

cheese, pineapple, ham

ham, cheese, pineapple

ham, pineapple, cheese

pineapple, cheese, ham

and

pineapple, ham, cheese

as separate outcomes, when in reality, they are all the same thing.

To get the answer we want we just divide by 3! or 6. 

We divide by 6 because (as we saw above) any combination of three outcomes has 6 different permutations (because 3!=6)

We can find the number of combinations using...

Remember 

Hope this is helpful!!!