Combination Calculator

This combination calculator (n choose k calculator) is a tool that assists you in figuring out the number of combinations in a set (commonly denoted as nCr) as well as in displaying every single combination (permutation) of your set, up to the length of 20 elements.

Be cautious, though! Such lengthy phrases could take our combination generator several seconds to discover. Try our combination calculator right away if you're curious about the range of possible combinations that may be created from a given set of components and sample size.

Compute combinations

If you still need to figure out what a combo is, the following article will explain everything. Here, you'll discover the combination formula and a definition for combinations (with and without repetitions).

You'll learn how to compute combinations as well as what the probability and linear combinations are. Finally, we'll discuss how permutation and combination relate to one another. In a nutshell, permutation considers the order of the components, whereas combination does not. More details are provided below.

Have you ever wondered how likely it is that you'll take home the lottery's top prize? How likely is it that I'll take second place? You must utilize combinations to respond to both questions as well as comparable ones. We have a unique tool only for issues of that nature. Our lottery calculator offers a lottery formula in addition to estimating the combo probability of winning any lottery game.

Try it! In fact, you'll learn how large (or little) those figures are. You might also be interested in scientific notation, a practical method for expressing really large numbers. For instance, you may write 145,000,000,000 as 1.45 1011 and 0.000000643 as 6.43 10-7. Isn't that easier? Check out the guidelines for a scientific notation for further details.

What is a combination? - combination definition

According to the definition of a combination, it is the number of ways in which r items may be selected from a collection of n different objects (thus, "n pick r" issues). Contrary to permutation, the order in which you pick the components is not crucial (you can find an extensive explanation of that problem in the permutation and combination section).

It is only possible to find all possible combinations of a collection of things using mathematics. You've probably previously learned how to do things like discover the greatest common factor (GCF) or the least common multiple (LCM). A combination, however, is a completely different matter. Let's examine its potential complexity.

Consider a bag containing twelve coloured balls, each one in a distinct hue. Five balls are chosen at random. How many different ball sets can you get? Or, to put it another way, how many possible combinations are there?

How to calculate combinations? - combination formula

Combination formula: how to compute combinations Mathematicians offer the precise solution to a wide range of issues, such as how to compute volume or square footage. Is it possible to estimate the quantity of combinations using balls as in the previous example?

Fortunately, you don't have to list every potential set. Then, how do you calculate the combinations? You may quickly ascertain the number of combinations by using the combination formula shown below:

Where:

The number of combinations is given by the formula C(n,r), where n is the total number of items in the set and r is the number of components you select from it.

A factorial is represented by the exclamation symbol! For additional details on this subject, see our factorial calculator. The binomial coefficient is another name for the expression on the right. It is also used in our binomial distribution calculator, which is another statistical calculator. The calculations on this website have certain similarities with others; for instance, the binomial calculator uses the nCr calculator.

Let's use this equation to solve the issue we have with the colored balls. We need to count the number of possible combinations:

C(12,5) = 12!/(5! * (12-5)!) = 12!/(5! * 7!) = 792.

Our NCR calculator allows you to confirm the outcome. The whole list of combinations will be included. Be warned though, there are already quite a few choices to display (792 to be exact). We restricted our combination generator to a certain, maximum amount of possibilities to prevent a situation where there are too many created combinations (2000 by default). Anytime you wish, you may alter it in advanced mode.

More about calculating combinations

You might have noticed that the number of combinations for selecting only one element is n, as calculated using the combinations formula. On the other hand, there is only one method to choose all of the components. Let's use our example to test this combo characteristic.

You have a total of 12 things, which is equal to n. The NCR calculator shows each letter as a different color of the ball, such as A being red, B being yellow, C being green, and so on. There are 12 distinct balls, hence there are 12 options if you pick only one element from that set with r = 1 at once. But if you select r = 12 items, only one combination that includes each ball will be available. Make your own attempt with the n pick r calculator!

You have most likely learned everything there is to know about combinations and the combination formula by this point. If that's not enough, we go into more detail in the following sections about the distinctions between permutation and combination (which are frequently mistakenly thought of as the same thing), combination probability, and linear combination.

Permutation and combination

You might be curious about when permutation is preferable to combination. It all depends on when to consider ordering. For illustration, suppose you had a deck of nine playing cards with the numerals 1 through 9. You choose three cards at random, arrange them in a line on the table, and come up with a three-digit number, such as 425 or 837. Can you make as many different numbers as possible?

P(9,3) = 9!/(9-3)! = 9!/6! = 504

Make sure of the outcome using our NCR calculator! How many different possible combinations exist?

C(9,3) = 9!/(3! * (9-3)!) = 9!/(3! * 6!) = 84

There are fewer always combinations than there are permutations. This time, it is six times smaller (84 times 3! equals 6 for a result of 504). It results from the fact that, like in the preceding example with the three colored balls, each set of three cards can be rearranged in six distinct ways.

In many academic disciplines, combination and permutation are equally important. They appear in physics, statistics, economics, and, of course, mathematics. We have more useful technologies that may be applied in similar circumstances. Try out the significant figures calculator, which explains the laws of significant figures and allows you to rapidly estimate a logarithm with whatever base you choose. It also explains what "significant" numbers are.

Permutation and combination with repetition. Combination Generator

We must introduce a comparable selection with permitted repeats in order to round up our discussion of permutation and combination. This means that after selecting an element from the set of n different objects, you always return it to that set.

In the case of the example with the various colored balls, you pick one ball out of the bag, keep track of which one you drew, and then place it back in the bag. Comparatively, in the second card example, you choose one card, note the number on that card, and then reshuffle the deck. In this method, you may, for instance, have a permutation of 228 or a combination of, say, two red balls.

You presumably predicted that both formulas would become rather difficult. Even still, it's not as complicated as figuring out how much alcohol your homebrewed beer has (which, by the way, you can do with our ABV calculator). In fact, the equation becomes considerably simpler in the case of permutation. Combination with repetition follows the following formula:

And for repetition and permutation:

P'(n,r) = nr.

The summary of the distinctions between the four methods of selecting an object—combination, combination with repetition, permutation, and permutation with repetition—is shown in the image below. You have four balls of different colors, and you select three of them as an example. You can choose one of the balls more than once in choices with repetition. Be cautious when experimenting with the permutations because there will be a lot of different sets! However, you can still calculate how many of them there are in a secure manner.

Combination probability and linear combination

Let's start with combination probability, which is a crucial component of many statistical puzzles (we have a probability calculator that explains it everything). It should be clear from the example in the image above: you choose three of the four coloured balls from the bag.

Let's imagine you want to know the likelihood (probability) that one of them will be red. The red ball appears in three of the four distinct combinations. Therefore, the combined probability is:

Pr = 3/4 = 75%.

75% of the time, if you select three balls at random from the bag, you'll choose a red ball. The % symbol is typically used to denote likelihood. If necessary, you may find out how to find percentages in our other calculator.

Let's say you choose one ball, note the color you received, and then put it back in the bag. What is the likelihood that at least one red ball will appear in the combination? This issue involves "combination with repetition." Given that the red ball appears in 10 of the possible twenty combinations shown in the image above:

Pr = 10/20 = 50%.

Are you surprised by that? It shouldn't be, after all. When you return the first ball, for example, the blue ball, you can also draw a second and third ball. As a result, the chances of receiving a red ball are reduced. Analogous considerations can be performed with permutation. Utilize the colored ball bag to attempt to solve a puzzle.

what is the probability that your first picked ball is red?

Say you want to verify it for yourself since you don't believe us. You draw three of the four balls and look to see if there is a red ball (like in the first example of this section). If you go through that process three more times, only one out of every four times (or 25% of the time) will you acquire the red ball. According to theory, you anticipated 75%.

What took place? This is how probability operates, then. The outcome of repeating an experiment several times is described by the law of big numbers. If you sketch something 100 times, for example, you'll be a lot closer to 75%.

Furthermore, the law of large numbers almost always produces the conventional normal distribution, which can be used to describe things like a person's height or IQ with a p-value. We describe how to calculate the p-value using the z-score table in the p-value calculator. Despite how difficult it may seem, this isn't really that difficult.

Do you know what a linear combination is? In spite of the word combination, it doesn't actually have many similarities with what we have learned so far. We'll nonetheless make an effort to do so succinctly.

A linear combination is created by multiplying a collection of terms by a constant, then adding the results. Because of the de Broglie equation, it is extensively employed in wave physics to predict the diffraction grating equation and even in quantum physics. Here are a few typical illustrations of a linear combination:

examples of linear combination

• Vectors. Each 3D vector may be broken down into its three component unit vectors, e1 = (1,0,0), e2 = (0,1,0), and e3 = (0,0,1). For instance, the linear combination v = (2,5,3) = 2e1 + 5e2 + 3e3 is given.
• Functions. Consider that you have two functions, f(x) = ex and g(x) = ex. You may construct linear combinations that depict the hyperbolic sine sinh(x) = f(x)/2 - g(x)/2 or cosh(x) = f(x)/2 + g(x)/2 from those two functions. With the standard sine and cosine, you may do the same task, but you must employ the fictitious integer i. In the square root calculator's last section, we go into greater detail about it.
• Polynomials. Say you wish to represent the function q(x) = 2x2 + x + 3 as a linear combination of the three polynomials p1(x) = 1, p2(x) = 3x + 3, and p3(x) = x2 - x + 1. Although it isn't always feasible, in this instance, q(x) = -2p1(x) + p2(x) + 2p3 (x).

What distinguishes combination from permutation?

If we care about the order of the components, then combinations and permutations vary fundamentally from each other in math:

• We put the things in sequential order since the sequence is important in permutation.
• We choose a group of things from a bigger collection since the order is irrelevant in combinations.

How can I compute combinations into permutations?

If you already have a combination and want to create a permutation out of it, you must impose order on the elements in your set by selecting one of the various orderings. The number of permutations of r items selected from n items is therefore equal to r!, where r! is the number of orderings of these r things, multiplied by the number of combinations of r items selected from n items.

Using permutations, how can I compute combinations?

You must eliminate order, or treat all potential reorderings as the same object, in order to convert a permutation that already exists into a combination. The quantity of combinations of r things selected from n items is therefore equal to the quantity of permutations of r items selected from n items divided by the quantity of orderings of these r items, i.e., by r!

What combinations are there for a seven-letter word?

If a word includes seven letters, you can arrange them in 7! = 5040 different ways (simple permutations of seven items). However, the number of arrangements decreases if some letters occur more than once! For illustration:

The answer is 2520 if the word is "WITNESS," which has the letter "S" present twice.

If the word is "someone," we divide 7 by 2 because "O" and "E" occur twice. 1260 is the outcome when 2! * 2! Equals 4.

If the word is "unknown," we have "n" three times; thus, we divide 7 by 3 to get 6, which gives us 840.