• Solve counting problems using the Addition and Multiplication Principle
• Solve counting problems using permutations involving [latex]n[/latex] distinct objects
• Solve counting problems using combinations
• Find the number of subsets of a given set
• Solve counting problems using permutations involving n non-distinct objects

Application Problems Involving Counting

When solving counting problems, it’s important to first understand the structure of the problem and identify what exactly is being counted.

Here are a few key things to pay attention to:

• Addition Principle: Use addition when you have separate, mutually exclusive choices or events. For example, if you can choose either one of several appetizers or one of several desserts, you add the number of options.
• Multiplication Principle: Use multiplication when you have multiple stages or parts of a process, where each choice or event is independent of the others. For instance, if you choose an appetizer, an entrée, and a dessert, you multiply the number of options for each to find the total number of possible meal combinations.
• Permutation: Use permutations when the order of selection matters. For example, if you are arranging books on a shelf, the order in which the books are placed is important, so you would use permutations.
• Combination: Use combinations when the order of selection does not matter. For example, if you are selecting members for a committee, the order in which they are selected is irrelevant, so you would use combinations.

By carefully analyzing the problem and understanding these principles, you can decide the appropriate method to use—whether it’s addition, multiplication, permutations, or combinations—to find the correct count.

Let’s play the lottery! First assume that you not only need to pick six specific numbers from [latex]1 – 49[/latex], but you need to pick them in the correct order.  If this is the case, you know you need to use a permutation to figure out the size of the sample space.

[latex]P\left(n,r\right)=\dfrac{n!}{\left(n-r\right)!}[/latex]

In this case, [latex]n[/latex] is the possible numbers, which is [latex]49[/latex], and [latex]r[/latex] is the number of choices you make, which is [latex]6[/latex].

[latex]P\left(49,6\right)=\dfrac{49!}{\left(49-6\right)!}[/latex]

[latex]P\left(49,6\right)=\dfrac{49!}{43!}=10,068,347,520[/latex]

This tells you that there is one way out of about [latex]10[/latex] billion to win.  Your chances are not good at all.

Fortunately, most lottery winnings do not depend on order so you can use a combination instead.

[latex]C\left(n,r\right)=\dfrac{n!}{r!\left(n-r\right)!}[/latex]

[latex]C\left(49,6\right)=\dfrac{49!}{6!\left(49-6\right)!}[/latex]

[latex]C\left(49,6\right)=\dfrac{49!}{6!\left(43\right)!}[/latex]

[latex]C\left(49,6\right)=\dfrac{49!}{6!\left(43\right)!}=13,983,816[/latex]

Notice that the sample space has been greatly reduced from about [latex]10[/latex] billion to about [latex]14[/latex] million.  So the likelihood of you winning is much greater than before, but still very slim.

After seeing how counting principles work in games of chance like the lottery, let’s explore how these same concepts apply to decisions you make every day on campus – from selecting your class schedule to joining student organizations.