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The Fundamental Counting Principle Solutions

• The Fundamental Counting Principle (FCP) is used to determine the    number of options    (or arrangements) when there are    several independent choices   .
• If one choice has n options and another has m options, then nm (the product) is the    total number of choices   .
• There are many different types of scenarios that    impact how you apply    the principle.
• With decreasing options, if you have a set group of items and select them one at a time, you will    subtract 1 from the remaining number of options   .  
• Each selection depends on    what was selected previously   .  
  $\begin{array}{cccccc}\boxed{\underline{ 6 }}& \underline{\boxed{ 5 }}& \underline{\boxed{ 4 }}& \underline{\boxed{ 3 }}& \underline{\boxed{ 2 }}& \underline{\boxed{ 1 }}\\ \left(n\right)& \left(n–1\right)& \left(n–2\right)& \left(n–3\right)& \left(...\right)& \left(1\right)\end{array}=720$
• A factorial (!) is a mathematical shorthand used to represent    multiplication from n to 1   . 
• It can be used to determine the    number of possible arrangements    of n-elements.
• Because you can arrange nothing into one group of nothing $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{0}\mathbf{!}\mathbf{=}\mathbf{1}\mathbf{ }\mathbf{ }\mathbf{ }}$.

Example 1

Mina only buys clothing that pairs well together. She has 7 tops, 4 bottoms, 3 pairs of shoes, and 3 bags. How many different outfits is Mina able to create?

$$\begin{gathered} \begin{array}{cccc}\underline{ 7 }& \underline{ 4 }& \underline{ 3 }& \underline{ 3 }\\ \mathrm{tops}& \mathrm{bottoms}& \mathrm{shoes}& \mathrm{bags}\end{array}=252 \\ 7·4·3·3 \end{gathered}$$

Mina can create    252 unique    outfits with her current options.

Example 2

Example 3

At the quick checkout lane, Marion removed the eight different items from the basket one at a time and scanned them before paying. How many arrangements are there when scanning the items?

$n=8$

$8!=8·7·6·5·4·3·2·1=40,320$

There are 40,320 possible ways to scan the eight items. 

Note

Once something is scanned, you have one less item than you started with.