Combinatorics: Formulas & Examples

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  • 0:00 Combinatorics
  • 0:59 Typical Combinatoric…
  • 2:30 Counting the Possibilities
  • 6:16 Looking at More Examples
  • 7:18 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

The mathematical field of combinatorics involves determining the number of possible choices for a subset. In this lesson, we use examples to explore the formulas that describe four combinatoric cases based on ordering and repetition.

Combinatorics

It's your dream job to create recipes. Well, maybe not. But if you wanted to explore variations of ingredients in a recipe, you might start with three basic spices: cumin, oregano, and basil. How many recipe variations result choosing two out of these three spices? What if we decide to add one spice at the beginning and a second at the end of the cooking? How many recipe variations? What if we can use the same spice at the beginning and at the end? Again, how many recipe variations are we considering?

We could list all these possible recipes for all of these cases because the number is small. But what if we have to consider a selection of spices from a larger number of spices. Eventually, it wouldn't be practical to list all the possible recipes and to count them. That's where the field of combinatorics comes in. In this lesson, we will explore formulas using examples. But first, some background math.

Typical Combinatoric Calculations

The factorial is expressed as n!. We read this as: n factorial.

Some facts about the factorical include:


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For example:


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The factorial will appear in our calculations. Here is one of those formulas:


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We read the left-hand-side as: n take k. Let's say that n is 5 and k is 2. Then, we get the formula with the values plugged in that you see below and that eventually equals 10.


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By the way, it is also true that:


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We can show this in the formula below that says:


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Our last type of math calculation is expressed with double parentheses, as you can see here:


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Here's an example of this playing out, which you can see, ultimately, comes to equal 15.


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We can now explore how to use this math for counting possibilities.

Counting the Possibilities

Before we start using the formulas, let's actually count the possibilities for our recipe situation. Then we can use the formulas and verify the results.

For conciseness, we will use C for cumin, O for oregano and B for basil.

When determining the number of possible choices when we select k from n, we first decide if the order matters or not.

Let's say that we have those three spices and can choose any two for the recipe. One spice is added at the beginning of the cooking process. The other spice is added at the end. Here, the order matters.

Now, we decide if repetition is allowed. Let's say that there is no repetition of the same variable (which is spice in our case). This is called a permutation with repetition not allowed. The formula is given by:


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Lets actually list the possible recipes: (C,O), (O,C), (C,B), (B,C), (O,B), (B,O). There are six possible recipes when choosing 2 out of 3 where the order matters and there is no repetition.

What does the formula give us? As we can see, it turns into:


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We can think of this as having 3 choices for the first spice and 2 choices for the second spice. This gives 3 x 2 = 6 possible recipes.

A permutation with repetitions allowed has the formula:


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In the recipe example, permutations with repetitions could happen if you can use the same spice at the beginning and at the end. The list of spice combinations is now larger: (C,C), (O,O), (B,B), (C,O), (O,C), (C,B), (B,C), (O,B), (B,O). There are 9 possible recipes.

Our formula for permutations with repetitions allowed would be:


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We can think of this situation as having 3 choices for the first spice and 3 choices for the second spice. Then, 3 x 3 = 9 possible recipes.

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