## Introduction

A combination calculator often denoted as nCr, is a valuable tool to determine the number of combinations in a set. This tool not only provides the result but also lists every possible combination or permutation of a setup to the length of 20 elements. In this article, we’ll explore the definition of combinations, the combination formula, and practical examples of its application.

## What is a Combination?

A combination is the number of ways in which you can choose ‘r’ elements out of a set containing ‘n’ distinct objects. Unlike permutations, the order in which you choose the elements is not essential. For example, if you have a bag with twelve different colored balls and pick five at random, the number of distinct sets or combinations can be calculated.

## Combination Formula

Mathematically, combinations are calculated using the formula:

*C*(*n*,*r*)=*r*!(*n*−*r*)!*n*!

Where:

*C*(*n*,*r*) is the number of combinations.*n*is the total number of elements in the set.*r*is the number of elements chosen from this set.*n*! denotes the factorial of*n*, representing the product of all positive integers up to*n*.

## Example Calculation

Let’s apply the combination formula to the example with twelve colored balls, where we want to choose five:

*C*(12,5)=5!(12−5)!12!=5!⋅7!12!=792

This means there are 792 different combinations of choosing five balls from the set of twelve.

## Permutation and Combination

Permutation takes into account the order of members, while combination does not. For instance, if you pick five balls at random from a bag, the order in which you pick them matters in permutations, but not in combinations.

## Permutation Formula

The permutation formula is given by:

*P*(*n*,*r*)=(*n*−*r*)!*n*!

If you know the number of combinations, you can easily calculate the number of permutations:

*P*(*n*,*r*)=*C*(*n*,*r*)⋅*r*!

## Permutation vs. Combination Example

Consider choosing three cards from a deck of nine cards with digits from 1 to 9. If the order matters, the number of permutations is 504, whereas the number of combinations is 84.

## Combination with Repetition and Permutation with Repetition

In some cases, repetitions are allowed when selecting elements from a set. The formulas for combination with repetition and permutation with repetition are more complex but follow a similar pattern.

## Combination Probability and Linear Combination

Combination probability is crucial in statistical problems. It represents the likelihood of certain combinations occurring. For example, if you pick three out of four colorful balls from a bag, the combination probability of getting a red ball might be 75%.

Linear combination, on the other hand, involves multiplying each term in a set by a constant and adding the results. It is often used in physics to describe phenomena like wave physics and quantum physics.

## Conclusion

Understanding combinations, permutations, and their applications is essential in various mathematical and statistical problems. Whether you’re dealing with colorful balls in a bag or cards from a deck, the combination calculator and formulas provided here can help you navigate and solve these problems effectively.