# Listing outcomes

## In a nutshell

To find the probabilities of outcomes in an experiment, you need to know how many different outcomes there are. In some cases, you may have to list all possible outcomes, and in other situations you can use the multiplication principle to work out how many there are.

## Combinations

You can use combinations to find the total number of possible events of an experiment.

##### Example 1

*Nina bought $3$ different cake mixes: chocolate, strawberry and lemon. She also bought chocolate chips and pink hearts to decorate them, and for any cake she must have $1$ mix and $1$ decoration. The combinations she can make can be listed as follows:*

- Chocolate mix with chocolate chips

- Strawberry mix with chocolate chips
- Lemon mix with chocolate chips
- Chocolate mix with pink hearts
- Strawberry mix with pink hearts
- Lemon mix with pink hearts

**Tip: **Do your best to list the options in a logical order, so you don't miss one by accident!

## Listing possible combinations

#### Procedure

1. | Make a table where the column headings detail the different options (for the cake example this is $2$: the cake mix and the decoration). |

2. | Fill in the table with a combination of options, writing the relevant option in the correct column. |

3. | Repeat until all combinations are listed. |

##### Example 2

*Nina can list the possible combinations of her cakes in the following table:*

#### Number | #### Cake Decoration | #### Cake Mix |

$1$ | Chocolate chips | Chocolate |

$2$ | Chocolate chips | Strawberry |

$3$ | Chocolate chips | Lemon |

$4$ | Pink hearts | Chocolate |

$5$ | Pink hearts | Strawberry |

$6$ | Pink hearts | Lemon |

## Number of combinations

It is possible to calculate the number of possible combinations without listing them all.

#### Procedure

1. | Look at the different aspects of each combination available. |

2. | Determine the number of choices for each aspect. |

3. | Multiply the number of choices for every aspect together |

##### Example 3

*Luke has $3$ pairs of trousers, $4$ shirts and $2$ pairs of shoes. How many different outfits can he wear?*

*Total combinations: Number of trousers $\times$ Number of shirts ** $\times$ Number of shoes *

*$3\times4\times2 = \underline{24}$*