Combinations Formula & ExamplesUsing technology:What are Derangements?

Permutation and Combination: Definition

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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:

Lottery organizations need lớn know how many ways numbers can be chosen in order to lớn calculate odds.Slot machine manufacturers need to know how many ways the pictures on the wheels can line up, to lớn calculate odds and prize money.

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:


However, 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 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:

Combination locks can have any number in any position (for example, 9, 8, 9, 2), so repetitions are allowed. The number “9” appears twice here.Lottery numbers don’t allow repetition. The same number won’t appear twice in the same ticket. For example, you can pick numbers 67, 76, & 99. But you can’t choose 67, 67, & 67 as your winning ticket.

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.

N is the number of things you are choosing from,r is the number of items.

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.

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

N is the number of things you are choosing from,r is the number of items.

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?