**Combinations Formula & Examples**

**Using technology**:What are

**Derangements**?

## Permutation and Combination: Definition

Watch the video for an overview and examples:

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A **permutation **or **combination **is a set of ordered things. The “things” can be anything at all: a danh sách of planets, a mix of numbers, or a grocery list. The menu can be in a phối order (like 1st, 2nd, 3rd…) or a list that doesn’t have khổng lồ be in order (like the ingredients in a mixed salad). The hardest part about solving permutation and combination problems is: **which is a combination & which is a permutation? **

**Permutation**: If you

*do*care about order, it’s a permutation. Picking winners for a first, second & third place raffle is a permutation, because the order matters. Permutation isn’t a word you use in everyday language. It’s the more complex of the two. Details matters: Eggs first? Then salt? Or flour first?

The word “combinations” has slipped into English usage for things like a “combination lock”. However, the kind of lock you put around your bicycle *should *(at least, from a stats point of view) be called a “permutation lock,” because the order *does* matter.

## Use in Real Life

Why do we care about all this in real life? **Combinations & permutations have hundreds (possibly, thousands) of applications,** the most obvious of which is gambling:

## Permutations = More Possibilities

You always have fewer combinations than permutations, và here’s why:Take the numbers 1, 2, 3, 4. If you want to know how many ways you can select 3 items where the order doesn’t matter (and the items aren’t allowed to repeat), you can pick:

123234134124However, if you want permutations (where the order *does* matter, the same set has 24 different possibilities. Just take the first set of numbers listed above 1, 2, 3 and think of the ways you can order it.

There are six ways lớn order the numbers, which means there are 4 x 6 ways khổng lồ order the mix of four numbers.

## Repetitions

Repetitions are just repeating numbers. They become important when it comes lớn choosing the right formula.

123 has no repetitions (each of those numbers is unique).223 has the number 2 repeated.Allowing **repetition **depends on your situation. For example:

Logic should tell you if repetitions are allowed. For example, if you’re dealing with items that aren’t going lớn be replaced (like lottery balls), then you’re looking at no repetitions allowed.Back to Top

## The Formulas

Combination (C) và permutation (P) each have their own formula:

This is just multiplication & division. The “!” is the factorial symbol. That’s just a special way of multiplying numbers. Lớn get a factorial, multiply the number by each number below it until you get to lớn 1. For example:4! = 4 x 3 x 2 x 1 = 242! = 2 x 1 = 2Google can work out factorials for you. Type*4!*into a Google tìm kiếm and you’ll get the answer (24).

## Permutation Formulas

There are two permutation formulas. Which one you choose depends on whether you have repetitions.

**For repetitions**, the formula is:**nr. **

For example, let’s say you are choosing 3 numbers for a combination lock that has 10 numbers (0 lớn 9). Your permutations would be 10r = 1,000.

Xem thêm: Mẫu Lời Cảm Ơn Quý Thầy Cô Và Các Bạn Đã Lắng Nghe, Mẫu Lời Cảm Ơn Trong Slide Powerpoint

**For NO repetitions**, the formula is:**n! / (n – r)!**

For example, let’s say you have 16 people to pick from for a 3-person committee. The number of possible permutations is:16! / (16 – 3)! = 16! / 13! = 3,360.Back khổng lồ Top

## Combinations Formula & Examples

The combinations formula is:

## Example 1: 5 Choose 3

5C3 or 5 choose 3 refers to how many combinations are possible from 5 items, taken 3 at a time. What is a combination? Just the number of ways you can choose items from a list. For example, if you had a box of five different kinds of fruit & you could choose 2, you might get an apple và an orange, an orange and a pear, or a pear và an orange. But how many combinations are possible?