Combination Calculator

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C (n, r) = n! (r! × (n - r)!)


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Combination Calculator 

Our combination calculator determines the total number of possible combinations that can be acquired from a sample that is taken from a population or bigger set.  In essence, you can estimate how many combinations you can get from a subset representing a bigger and generalized set. This possibility calculator is your best shot to calculate your answer if you hate math.

What is Combination

It is the number of ways to choose a sample of ‘r’ elements from a set of ‘n’ distinguished objects where order is irrelevant and replacements are not permissible.

Difference between Combinations and Permutations?

The combination is the way things can be arranged together. For instance, consider 3 balloons: red, white and green. There are 9 types of groupings in which these balloons can be arranged without regard for any type of order unlike in permutations where the order is key. 

For permutations, consider a typical combination lock that unlocks only when 3 digits are positioned with each other in a particular order. Consider this, you have a digital lock with the passcode: 3,2,4.

Now, if someone sneaks in and finds your door locked, it’s quite unlikely that they would be able to open it by brute-forcing the digits 3,2 and 4 in some random order like 4,3,2 unless, they get these digits in exactly the right order, that is 3,2,4 which is unlikely, although possible in a brute force attack because the order 3,2,4 does exist within all possible combinations. So, as is clear from the preceding discussion, combinations don’t care about the order, permutations do. 

You can use online tools like number combination generator to generate the total number of groupings from a random set of elements.

nCr formula

You can determine the number of possible groupings with the ncr formula: It has been stated below.

\(C(n,r) = \dfrac{n!}{(r! \times (n-r)!)}\)


C (n,r): is the total number of combinations

n: total number of elements in the given set

r: number of elements chosen from the set for sampling

!: factorial 

How to calculate combination

To calculate combination, all you need is the formula, that too, in case you want to determine it manually. In this section, we will show you how it’s done.

To give you a practical illustration, we would be calculating the number of possible combinations in two different configurations of sets and subsets. Namely, the 3 choose 2 and 4 choose 4.

3 choose 2 

3 choose 2 basically means choosing a number of sampling elements ‘r’ from the total number of elements in the given set ‘n’. In this method. In other words, 3 choose 2 means representing all possible configuration for 3 elements in a combination of 2 elements at a time. 

For instance, if you come across a box of 3 different types of fruit and you could choose only 2, you might get either an apple and an orange, or an orange and a Pineapple, or an Apple and a Pineapple. But can you know how many pairs are possible? Let’s do the math and find out. 

To calculate combinations:

\(C(n,r) = \dfrac{n!}{(r! \times (n-r)!)}\)

\(C = ?\)

We know,

\(C (3,2) = \dfrac{3!}{2!}\times (3-2)!\)

\(= \dfrac{3!}{2!} \times (1)!\)

Multiplying each value with respective factorial: (factorial preceding a number equals the multiplication of the number with all the previous numbers that precede it)

\(= \dfrac{3\times2\times1}{2\times1}\times1\times1 \)  

\(= \dfrac{6}{2}\)

\(= 3\)

4 choose 4

Now going for, \(4C4\):

We apply the same methodology and the same formula.

\(C(n,r) = \dfrac{n!}{(r! \times (n-r)!)}\)

\(C =?\)

\(n = 4 , r = 4\)

\(C(4,4) = \dfrac{4!}{(4! * (4-4)!)}\)

Applying factorial,

\(\dfrac{4\times3\times2\times1}{(4\times3\times2\times1 \times (0)!)}\)

\(=\dfrac{24}{24} \times 1\) 

\(= \dfrac{24}{24}\)

\(= 1\)

Combinations with and without repetition

Out there, you’d read about the headline above in very difficult mathematical jargon. That’s not the case here, we have tried to present this phenomenon to you in the simplest way possible.

With Repetitions

The general figuring out of combinations is usually without order. However, in combinations with repetitions, elements r from n can be used more than once, increasing the likelihood of repetitions.

Without Repetitions

On the other hand, combinations without repetition do not allow repetition because elements that have been combined already, cannot be combined again, making sure that repetitions don’t occur.

These groupings without repetition of ‘n’ elements occupied ‘k’ in ‘k’ are the varying groups of ‘k’ elements that can be produced by the given ‘n’  elements so that two classes are distinct only if they contain different elements (It’s a cool way of saying, the order doesn’t matter). They can be written as "\(C^{n}_{k}\)".

How to use our Combinations Calculator

You can choose our ncr calculator to determine combinations. Its elegant user-interface allows ease calculation. All you need to do for your part is simply input the total number (n) of elements in the given set and total number ‘r’ of the elements you want your combinations in.

Adding these input variables, simply click on ‘calculate’ and there you go, the combination formula calculator would use the "\(C^{n}_{r}\)" formula and produce the output (answer) in real-time.  

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