Hi there. In this math post, I cover the concept of the fundamental counting principle from the math field of combinatorics. This principle is one of the most basic concepts from combinatorics and probability.

## How Many Combinations Are Possible?

The fundamental counting principle is a concept that helps with determine the number of combinations or outcomes. This principle helps quite a bit in the field of probability. The focus here is more on the obtaining the number of combinations instead of computing probabilities.

If there are `p`

choices for one thing and `q`

choices for the second thing then there are `p x q`

ways of choosing `p`

and `q`

. We can also extend this with more items. The general case would be something like `p_1 x p_2 x p_3 x ... x p_n`

for the number of combinations from `n`

items with their own number of choices.

This concept will make more sense through examples.

## Examples

**Example One - Ice Cream Cone**

Bobby wants an ice cream cone with just one scoop of ice cream. The ice cream cone comes in either a small, medium or large size. For a flavour Bobby can choose only one of strawberry, vanilla, chocolate, mint, green tea, cherry or mango.

There are 3 choices for the ice cream cone size and there are seven flavours to choose from. Multiplying 3 and 7 together gives 21 different ice cream choices. Here is a partial table.

Cone Size | Flavour |
---|---|

Small | Strawberry |

Medium | Strawberry |

Large | Strawberry |

Small | Vanilla |

Medium | Vanilla |

Large | Vanilla |

... | and so on |

**Example Two - Clothing Outfits**

Lukas has 7 pairs of pants, 12 shirts, 10 pairs of socks. How many possible outfits are there? If Lukas loses 2 pairs of socks, how many less outfits are there?

Multiplying 7, 12 and 10 together gives 840 different outfits.

If Lukas loses 2 pairs of socks we have 7 pairs of pants, 12 shirts and 8 pairs of socks. The number of different outfits now is `7 x 12 x 8 = 672`

. This is less than 840 by 168.

**Example Three - PC Build**

This example is a partial representation of the many choices you can have when it comes to selecting a PC build. Brian plans to build a personal desktop computer. He has to choose one from 10 motherboards, one from 5 CPU chips, one from twelve graphics card options, a solid state drive for storage from 20 options and one option from many 50 PC cases. Brian does not have to worry about RAM memory and the other items as his brother as some leftover parts.

How many PC builds can Brian make?

We multiply the number of choices from each category. This would be `10 x 5 x 12 x 20 x 50 = 600 000`

. In real life this number of 600, 00 is not realistic as you would have to check if the parts are compatible with each other.

Posted with STEMGeeks