Enter your objects (or the names of them), one per line in the box below, then click "Show me!" to see how many ways they can be arranged, and what those arrangements are. Show
Note: 8 items have a total of 40,320 different combinations. For the sake of output and server capacity, we cannot let you enter more than 8 items! This combination calculator (n choose k calculator) is a tool that helps you not only determine the number of combinations in a set (often denoted as nCr), but it also shows you every single possible combination (permutation) of your set, up to the length of 20 elements. However, be careful! It may take even a couple of seconds to find such long terms for our combination generator. If you wonder how many different combinations can be possibly made of a specific number of elements and sample size, try our combination calculator now! If you're still not sure what a combination is, it will all be explained in the following article. You'll find here a combination definition together with the combination formula (with and without repetitions). We'll show you how to calculate combinations, and what the linear combination and combination probability are. Finally, we will talk about the relation between permutation and combination. Briefly, permutation takes into account the order of the members and combination does not. You can find more information below! Have you ever wondered what your chances are of winning the main prize in a lottery? How probable is winning the second prize? To answer both and similar questions, you need to use combinations. We've got a special tool dedicated to that kind of problem. Our lottery calculator doesn't only estimate combination probability of winning any lottery game, but also provides a lottery formula. Try it! You'll find out how big (or small) those numbers are, in fact. You might also be interested in a convenient way for writing down very long numbers called scientific notation. For example, What is a combination?  combination definitionThe combination definition says that it is the number of ways in which you can choose r elements out of a set containing n distinct objects (that's why such problems are often called "n choose r" problems). The order in which you choose the elements is not essential as opposed to the permutation (you can find an extensive explanation of that problem in the permutation and combination section). Seeking for every combination of a set of objects is a purely mathematical problem. You probably have been already taught, say, how to find the greatest common factor (GCF) or how to find the least common multiple (LCM). Well, a combination is an entirely different story. Let's see how complicated it might be. Imagine a bag filled with twelve balls, where each one is a different color. You pick five balls at random. How many distinct sets of balls can you get? Or, in other words, how many different combinations can you get? How to calculate combinations?  combination formulaMathematicians provide the exact solution for many various problems, e.g., how to calculate square footage or how to calculate volume. Is there a similar approach in estimating the number of combinations in the above example with balls? Luckily, you don't have to write down all of the possible sets! How to calculate the combinations, then? You can use the following combination formula that will allow you to determine the number of combinations in no time: C(n,r) = n!/(r!(nr)!) ,where:
The exclamation mark Let's apply this equation to our problem with colorful balls. We need to determine how many different combinations are there:
You can check the result with our nCr calculator. It will list all possible combinations, too! However, be aware that 792 different combinations are already quite a lot to show. To avoid a situation where there are too many generated combinations, we limited this combination generator to a specific, maximum number of combinations (2000 by default). You can change it in the advanced mode whenever you want. You may notice that, according to the combinations formula, the number of combinations for choosing only one element is simply By this point, you probably know everything you should know about combinations and the combination formula. If you still don't have enough, in the next sections, we write more about the differences between permutation and combination (that are often erroneously considered the same thing), combination probability, and linear combination. Permutation and combinationImagine you've got the same bag filled with colorful balls as in the example in the previous section. Again, you pick five balls at random, but this time, the order is important  it does matter whether you pick the red ball as first or third. Let's take a more straightforward example where you choose three balls called R(red), B(blue), G(green). There are six permutations of this set (the order of letters determines the order of the selected balls): RBG, RGB, BRG, BGR, GRB, GBR, and the combination definition says that there is only one combination! This is the crucial difference. By definition, a permutation is the act of rearrangement of all the members of a set into some sequence or order. However, in literature, we often generalize this concept, and we resign from the requirement of using all the elements in a given set. That's what makes permutation and combination so similar. This meaning of permutation determines the number of ways in which you can choose and arrange r elements out of a set containing n distinct objects. This is called rpermutations of n (sometimes called variations). The permutation formula is as below: 1.45 × 1011 6.Doesn't this equation look familiar to the combination formula? In fact, if you know the number of combinations, you can easily calculate the number of permutations: 1.45 × 1011 7.If you switch on the advanced mode of this combination calculator, you will be able to find the number of permutations. You may wonder when you should use permutation instead of a combination. Well, it depends on whether you need to take order into account or not. For example, let's say that you have a deck of nine cards with digits from 1 to 9. You draw three random cards and line them up on the table, creating a threedigit number, e.g., 425 or 837. How many distinct numbers can you create?
Check the result with our nCr calculator! And how many different combinations are there?
The number of combinations is always smaller than the number of permutations. This time, it is six times smaller (if you multiply 84 by Both combination and permutation are essential in many fields of learning. You can find them in physics, statistics, finances, and of course, math. We also have other handy tools that could be used in these areas. Try this log calculator that quickly estimate logarithm with any base you want and the significant figures calculator that tells you what are significant figures and explains the rules of significant figures. It is fundamental knowledge for every person that has a scientific soul. Permutation and combination with repetition. Combination generatorTo complete our considerations about permutation and combination, we have to introduce a similar selection, but this time with allowed repetitions. It means that every time after you pick an element from the set of n distinct objects, you put it back to that set. In the example with the colorful balls, you take one ball from the bag, remember which one you drew, and put it back to the bag. Analogically, in the second example with cards, you select one card, write down the number on that card, and put it back to the deck. In that way, you can have, e.g., two red balls in your combination or 228 as your permutation. You probably guess that both formulas will get much complicated. Still, it's not as sophisticated as calculating the alcohol content of your homebrew beer (which, by the way, you can do with our ABV calculator). In fact, in the case of permutation, the equation gets even more straightforward. The formula for combination with repetition is as follows:
and for permutation with repetition: 0.000000643 2.In the picture below, we present a summary of the differences between four types of selection of an object: combination, combination with repetition, permutation, and permutation with repetition. It's an example in which you have four balls of various colors, and you choose three of them. In the case of selections with repetition, you can pick one of the balls several times. If you want to try with the permutations, be careful, there'll be thousands of different sets! However, you can still safely calculate how many of them are there (permutations are in the advanced mode). Combination probability and linear combinationLet's start with the combination probability, an essential in many statistical problems (we've got the probability calculator that is all about it). An example pictured above should explain it easily  you pick three out of four colorful balls from the bag. Let's say you want to know the chances (probability) that there'll be a red ball among them. There are four different combinations, and the red ball is in the three of them. The combination probability is then: 0.000000643 3.If you draw three random balls from the bag, in 75% of cases, you'll pick a red ball. To express probability, we usually use the percent sign. In our other calculator, you can learn how to find percentages if you need it. Now, let's suppose that you pick one ball, write down which color you got, and put it back in the bag. What's the combination probability that you'll get at least one red ball? This is a 'combination with repetition' problem. From the picture above, you can see that there are twenty combinations in total and red ball is in ten of them, so: 0.000000643 4.Is that a surprise for you? Well, it shouldn't be. When you return the first ball, e.g., blue ball, you can draw it as a second and third ball too. The chances of getting a red ball are thus lowered. You can do analogical considerations with permutation. Try to solve a problem with the bag of colorful balls: what is the probability that your first picked ball is red? Let's say you don't trust us, and you want to test it yourself. You draw three balls out of four, and you check whether there is a red ball or not (like in the first example of this section). You repeat that process three more times, and you get the red ball only in one of four cases  What's more, the law of large numbers almost always leads to the standard normal distribution which can describe, for example, intelligence or the height of people, with a socalled pvalue. In the pvalue calculator, we explain how to find the pvalue using the zscore table. This may sound very complicated, but it isn't that hard! Have you ever heard about the linear combination? In fact, despite it have the word combination, it doesn't have much in common with what we have learned so far. Nevertheless, we'll try to explain it briefly. A linear combination is the result of taking a set of terms and multiplying each term by a constant and adding the results. It is frequently used in wave physics to predict diffraction grating equation or even in quantum physics because of the de Broglie equation. Here, you can see some common examples of linear combination:
FAQWhat is the difference between permutation and combination?The fundamental difference between combinations and permutations in math is whether or not we care about the order of items:
How do I calculate permutations from combinations?If you already have a combination and want to turn it into a permutation, you need to impose order on the set of items, i.e., choose one of the possible orderings for your set. Hence, the number of permutations of How do I calculate combinations from permutations?If you already have a permutation and want to turn it into a combination, you need to remove order, i.e., regard all possible reorderings as the same object. Hence, the number of combinations of How many ways can I arrange a 7 letter word?If the word has seven distinct letters, you have How many ways can 7 digits be arranged?N=1st×2nd×3rd×4th×5th×6th×7th=9×10×10×10×10×10×10=9×106N=9000000There are 9000000 possible combinations for a 7digit number.
How many combinations of 7 numbers are there in the lottery?In other words, there are 5040 different ways that the 7 numbers you choose can be filled out on your lottery ticketif you choose your 7 numbers correctly, any of these ways will make a winning ticket.
How many ways can 7 digit numbers can be formed if repetition is not allowed?There are 9 digits you can have for the first digit (can't have 0). There are then 9 more for the second (all except the first one). Similarly there are 8 for the third, 7 for the 4th, etc… So the total would be 9*9*8*7*6*5*4 = 544,320 7 digit numbers without repeated digits.
How many different 7 digit numbers are there?How Many 7Digit Numbers are there in all? 7digit numbers start from 1000000 and end with 9999999. There are ninety lakh (90,00,000) 7digit numbers in all.
