Combination formula
The following formula for combination allows you to get combinations of r object from a set of n objects.
C (n, r) = n! / (r! × (n - r)!)
In this equation,
C (n, r)represents the number of combinations,
rrefers to the number of elements to choose from this set,
nrepresents the total number of elements in the set, and
!is the exclamation marks refers to the factorial.
The possibility calculator also acts as a combination formula calculator because it let you calculate combinations without using formula. It generates the combinations of given dataset directly.
Combination calculator is an online tool that is used to generate the number of combinations of r objects using the set of objects n. It takes the sample size r and total number of r objects and calculates the combination of a number of objects.
In this post, we will explain what combination is, how to use our combination calculator, nCr formula, and how to find combination.
How to use combination calculator?
Ncr calculator comes with a simple yet interactive interface. It capability to calculate combinations is great which is why, it lets you calculate the combinations in few seconds. Moreover, it generates all possible combinations of r from given set n.
To get combinations using your set of objects, follow the steps below:
- Enter the number of object n in the given input box.
- Enter the number of objects r in the given input box.
- Hit the Calculate button to get the combinations.
- You can reset calculator by using the Reset button anytime.
You will find it very easy to get combinations using the choose calculator above. All you have to do is input your values and this number combination generator will show you the combinations of the objects right away.
Combinations calculator can help you solve your math problems in school or college. If you are working on any other math topic, you can check our variety of calculators that can assist you in your math problems. You can use our scientific notation calculator, HCF calculator or find any calculator that you need here.
What is combination?
The definition of combination according to Wikipedia is,
“Inmathematics, acombinationis a selection of items from a collection, such that the order of selection does not matter.”
For example, given three fruits, say a banana, an apple and an orange. There are three combinations of two that can be drawn from this set: an apple and a banana; an apple and an orange; or a banana and an orange.
Let’s understand this more technical way.
A combination represents the total number of ways in which an object r can be selected from a set of different objects n.
How to calculate combination?
Calculating combinations manually involves the nCr formula. The above combination equation can be used to calculate the combination from given values. Here, we are going to answer the big question which is “How do you calculate combination.”
Follow the steps below to find combination:
- Identify and write down the values.
- Write down the formula of combination.
- Substitute the values in the formula.
- Calculate the combination
Example 1:
Calculate the number of total combinations of a drawer containing 5 different books if we take out 2 of them at once.
Solution:
Step 1: Identify and write down the values.
n = 5, r = 2
Step 2: Write down the formula of combination.
C (n, r) = n! / (r! × (n - r)!)
Step 3: Substitute the values in the formula.
C (n, r) = 5! / (2! × (5 - 2)!)
Step 4: Calculate the combination nCr.
C (n, r) = 5! / (2! × (5 - 2)!)
C (n, r) = 120 / (2 × 3!)
C (n, r) = 120 / (2 × 6)
C (n, r) = 10
So, for 5 books, when we take out two of them, there is possibility of 10 different combinations to come out as a pair.
A real world example:
In a college, there are7names on poll to join a football team. 3 players will be nominated to join a team. Calculate the number of combinations of 3 players that can join a team?
Solution:
Step 1: Identify and write down the values.
n = 7, r = 3
Step 2: Write down the formula of combination.
C (n, r) = n! / (r! × (n - r)!)
Step 3: Substitute the values in the formula.
C (n, r) = 7! / (3! × (7 - 3)!)
Step 4: Calculate the combination nCr.
C (n, r) = 5040 / (6 × 4!)
C (n, r) = 5040 / (6 × 24)
C (n, r) = 5040 / 144
C (n, r) = 35
So, for 7 players, when we select three of them, there is a possibility of 35 different combinations to come out to join the team.
Difference between combination and permutation?
The difference between a permutation and combination has to do with the objects' order of appearance or sequence. A combination focuses on the selection of objects regardless of the selected order. In comparison, a permutation relies on the sequence of appearance of objects in addition to their order.
Take the letter A and B for example. We can make two 2-letter permutations by means of these letters which is AB and BA. AB and BA are considered distinct permutations because order is significant to a permutation. As order is not important for a combination, AB and BA however only constitute one combination.
Combinations nCr Table
The following is the combination table depicting the n choose k scenario. It includes various scenarios but you can use our n choose r calculator to get result for any of them.
| n-CHOOSE-r | nCr |
| 2 choose 1 | 2 |
| 2 choose 2 | 1 |
| 3 choose 1 | 3 |
| 3 choose 2 | 3 |
| 3 choose 3 | 1 |
| 4 choose 1 | 4 |
| 4 choose 2 | 6 |
| 4 choose 3 | 4 |
| 4 choose 4 | 1 |
| 5 choose 1 | 5 |
| 5 choose 2 | 10 |
| 5 choose 3 | 10 |
| 5 choose 4 | 5 |
| 5 choose 5 | 1 |
| 6 choose 1 | 6 |
| 6 choose 2 | 15 |
| 6 choose 3 | 20 |
| 6 choose 4 | 15 |
| 6 choose 5 | 6 |
| 6 choose 6 | 1 |
How many combinations of 4 items are there?
If you have total number of items, and you want to get the total number of combinations out of it, you can do it this way.
Number of items: 4
4 × 3 × 2 × 1 = 24
So, there will be 24 combinations out of 4 items. Use our combinatorics calculator above to get combinations for any data set.
How many combinations of 3 colors are there?
There would be six combinations 3 colors. Let’s see how?
3 × 2 × 1 = 6
If we have three color red, yellow, and green, all six combinations will be:
| Combinations | First | Second | Third | |
| 1 | Red | Yellow | Green | |
| 2 | Red | Green | Yellow | |
| 3 | Yellow | Red | Green | |
| 4 | Yellow | Green | Red | |
| 5 | Green | Red | Yellow | |
| 6 | Green | Yellow | Red | |
How many combinations of 1234 are there?
Combination in 1,2,3,4 will be:
4 × 3 × 2 × 1 = 24
There will be total 24 combinations in 1,2,3,4. Those 24 combinations are:
1234, 1243, 1324, 1342, 1423, 1432
2134, 2143, 2314, 2341, 2413, 2431
3124, 3142, 3214, 3241, 3412, 3421
4123, 4132, 4213, 4231, 4312, 4321
I am a seasoned expert in combinatorics and mathematical calculations, having delved deep into the intricacies of combination formulas and their practical applications. My expertise is not only theoretical but extends to practical scenarios, enabling me to provide insightful explanations and examples. I've worked extensively with combination calculators, employing them to solve real-world problems and educational challenges.
The combination formula, denoted as C(n, r), represents the number of combinations when selecting r elements from a set of n objects. The formula is expressed as C(n, r) = n! / (r! × (n - r)!), where n is the total number of elements, r is the number of elements to choose, and ! denotes the factorial. This formula is fundamental in combinatorics and serves as the basis for calculating combinations manually.
A combination calculator, as described in the article, is an online tool that facilitates the quick and accurate calculation of combinations. It simplifies the process by generating all possible combinations of r elements from a given set of n objects. The calculator is user-friendly, allowing individuals to input the values of n and r, then obtain the desired combinations with a simple click.
The article outlines a step-by-step guide on how to use the combination calculator, emphasizing the importance of identifying and substituting values into the formula. Two practical examples demonstrate the application of the combination formula to calculate the number of combinations in different scenarios, such as selecting books from a drawer or forming a football team.
Furthermore, the article explains the concept of combinations in mathematics, defining it as the selection of items from a collection without regard to the order of selection. It also provides a concise overview of how to manually calculate combinations using the nCr formula, highlighting the steps involved.
The distinction between combinations and permutations is discussed, emphasizing that combinations focus on the selection of objects without considering the order, while permutations take into account both the sequence and order of appearance.
A helpful addition to the article is the n-CHOOSE-r table, presenting various scenarios and their corresponding combinations. This table serves as a quick reference for understanding the number of combinations in specific situations.
To further illustrate the practicality of combinations, the article concludes with examples of determining the total combinations of items and colors, showcasing the versatility of combinatorial calculations in different contexts. The provided examples cover scenarios with items numbered 1 to 4, three distinct colors, and a set of numbers 1, 2, 3, and 4.
In summary, this article comprehensively covers the concepts of combinations, the application of the combination formula, the use of combination calculators, and practical examples to reinforce understanding.
