Over the past few months, many of us have been playing around with the assignment generator we built at AssignU.com. We’ve been doing research, trying to let you know how and when to use it, and perhaps even to teach you how to use it yourself. In this set of blog posts, we’re going to look at a few different ways we played around with the assignment generator and thought of new ways to use it—in all areas of life, not just mathematics.
If you want combinatorics help, you have to know the definition and real-life applications of combinatorics. So, what is combinatorics? It can be described as a stream or chapter of mathematics that deals with finite discrete structures. If you have a good knowledge of permutation and combination, you would not have a problem understanding the basics of combinatorics.
In mathematics, the term “combinatorics” has been used for representing a larger subset of Discrete Mathematics. In computer science, this chapter of mathematics has been regarded as crucial. For developing various algorithms and programs, one should have a sound knowledge of combinatorics.
An Introduction to Combinatorics Help
For understanding the combinatorics meaning, it is not enough to learn the definition. You need to understand the real-life applications of the theory. Nevertheless, one should find various problems related to combinatorics and solve them properly. Solving the combinatorics problems will help you to gain a good concept of the subject. In this article, different aspects of combinatorics will be discussed. These aspects include formula, features, and applications of combinatorics.
Crucial Features of Combinatorics
For a proper and deep understanding of combinatorics, you need to look at the combinatorics examples. Before going to the examples, it is important to analyze the features. This section or genre of mathematics has many features. Some of the notable features are highlighted in the following section.
- Combinatorics can help to count the structures of the provided size, kind, and numbers.
- A combinational design comes into the picture when particular criteria need to be fulfilled. At the same time, the combinational design helps to analyze the elements of the criteria.
- Through the external combinatorics, you can identify the greatest, optimal, and smallest elements among the groups of various elements.
When algebra is applied for resolving combinatorial problems, such combinational structures are termed algebraic combinatorics. In various combinatorics algorithms, you can find the use of algebra.
The Basics of Combinatorics
If you want combinatorics examples with answers, you need to know the basics of combinatorics. How does it work? What are the basic rules involved in the various combinatorics problems? Many mathematics experts call combinatorics “The Fundamental Counting Principle”. This counting principle encourages people to multiply the number of each discrete outcome for determining the number of the overall outcomes.
For simplifying the theory, let us take an example in consideration. Assume that there are X options in one case and Y options in another case. So, how many cases you have together? The answer is the multiplication between X and Y. So, overall outcomes should be X*Y.
Let us now consider some real-life numbers for even simplified understanding. You went for having your lunch, and you get the following three features:
- There are five choices for burgers
- Seven choices for soda water
- Two choices for chips
So, how many options are there for you in the lunch? The answer is 5x7x2 = 70. This rule of mathematics should be applied when you are asked to specify the total number of outcomes from different choices. Make sure that the choices do not affect each other.
To understand combinatorics meaning, you need to understand the basics of permutations. A permutation has been regarded as the arrangement of various items. For understanding the topic, let us take the classic game Mario Kart 64 into consideration.
A permutation is the arrangement of the items. In Mario Kart 64, you can notice eight racers in the game. So, what is the number of total outcomes for a race? You need to put the things in certain orders, and each order will be unique. Using the Fundamental Counting Principle, you can get the answer to this problem. With eight racers, you can have eight possible slots to fulfill.
For the first position, you have a total of 8 choices. So, when the first slot is filled, you will be left with seven choices. Similarly, finishing the 2nd slot will leave you at six choices in hand. Once the third slot is fulfilled, you will have 5 choices. This is how it goes till the last slot.
In combinatorics problems, you will also notice the factorials. These factorials come into the picture because you have to multiple the options you have in hand. In every step, you keep losing an option, as mentioned above. For the “n” number of items, you will have a total n! outcomes. Permutation theory should be used, when you have to figure out the number of ways of arranging items. When arrange is the motto, you should apply the theory mentioned above.
Cases Where Order Is Not Important
When it comes to applications of combinatorics in real life, you will encounter cases where the order is not a factor. Imagine that there are 10 racers for a racing event. Now, which three racers are required to be featured on the racing manual? How many options do you have in such a scenario?
For answering the questions above, you do not need to consider the order. Here, the order is not an important factor. With 10 slots, you are starting with an 8! several arrangements. You have to pick three from the available arrangements randomly. As a result, five will remain left out. For resolving such combinatorics problems, you have to care for those who fall in what side of the brackets, instead of considering their orders.
Various Applications of Combinatorics
In the above section, you have learned what is permutation and combination with an example. Now, the question is why combinatorics is so important? How can it be applied in your daily life? Is there any real-life application of combinatorics at all? For combinatorics help, you need to know the answers to these questions. So, here are the answers for you:
- In computer molecular biology, combinatorics has been used.
- Extensive usage of combinatorics can be noted in various scientific researches.
- For database management theories and programs, combinatorics has a big role to play.
- Combinatorics is an important element of operation research.
- For pattern analysis, combinatorics has been used.
- In developing computer architecture, the use of combinatorics is noticeable.
So, you have learned the basics of combinatorics. Now, you have to solve various problems related to combinatorics.
How many combinations of K objects are there from a collection of n objects? The answer is this. It’s N factorial over N minus K factorial times K factorial. And to write that more quickly, we often use this notation called a binomial coefficient, which is defined as equal to that.
(n , k) = n! / [ (n-k)! k! ]
This is a pretty basic and common sort of counting problem, which is why its solution is so important to us.
Suppose we have five actors that are all auditioning to be in some play, and only three of these five actors will be selected. If we want to know how many different groups of actors are possible where we don’t care about the order in which they’re selected. We only care about the actors making up the group. That is a situation where we use combinations. It’s the number of combinations of three actors from a collection of five actors. So let’s go full speed ahead and see where this formula pops up in this situation. So we want to count the number of ways we can select three actors from a collection of five. The order doesn’t matter.
Our first steps into a solution might begin like this. We know three spots need to be filled by three actors that will be selected. Before any actors are selected, there are five possibilities for that first selection. For every five actors that could be selected first, four actors could be selected second. And for all of those 20 possibilities of actors that could be selected first, then second, three actors remaining could be selected third. So that’s a start. But that’s not exactly what we want. This number happens to be 20 times three, which is 60. That’s the number of ways that we could arrange three actors from a collection of five actors. But when I say arrange, I mean order, there is ordering in this count, and we don’t care about the order. So then we want to know how many orderings are being counted because we want to get rid of the alternate orderings.
Well, for any combination, that’s what we do care about the number of combinations. For any combination of three actors, let’s call them Bill, James, and Chad.
We want to count it just once for any combination of three actors like Bill, James, and Chad. But all of the alternate orderings are also being counted in this number 60. How many alternate orderings are there of three actors? Well, that would be three factorials. That’s the number of ways we can arrange three objects in order. Three factorials, of course, is equal to three times, two times one, which is six. We could have Bill, James, Chad or Bill, Chad, James or James, Chad, Bill or Chad, Bill, James or James, Bill, Chad or James, Chad, Bill. Those are our six possibilities. So what’s happening is that every combination of three actors that we care about is getting counted three factorial times instead of just ones, which is what we want.
So to correct our count, we have five times, four times three, the number of ways of ordering three actors from our collection of five. But then, since every combination we care about is being counted three factorial times because that’s a number of ways of ordering any selection of three actors, we need to divide by three factorial. This will give us a final count of sixty divided by three factorials, which is six, and that will be equal to ten. And that is the number of combinations of three objects from a collection of five.
And you might say that looks kind of similar to the formula we saw initially, but it certainly doesn’t look quite right. So how can we rewrite this in a form that looks more similar to the general formula? If we put five factorials in the numerator so that we’re trying to make something that looks more like this, then not only do we have to divide by three factorials, as we did before, to get rid of the alternate orderings of every combination of three actors. But we also have to divide out the number of orderings of actors that we’re not selecting. And since there are five actors and we are selecting three, the number of actors we aren’t selecting is five minus three. So the number of ways of ordering those actors we’re not selecting which are being counted here, we don’t want them to be counted, which is why we’re dividing them out. That number of ordering the actors we’re not selecting is five minus three factorial.
And now we see a form resembling the formula, which is the same thing we had above. We can see that pretty clearly. If I rewrite this a little bit on the numerator, we have that five factorial. And then in the denominator, we have five minus three factorial, which is two factorial, which is two times one. And then that’s getting multiplied by three factorials. And we see that two and one in the numerator and denominator cancel out. And we’re left with the same thing. We had above five times, four times three divided by three factorials. And that is where the formula comes from.
This (n, k) is called a binomial coefficient. It’s read “n choose k,” it’s the number of ways we can choose K objects from a collection of N objects where the order of selection doesn’t matter where only interested in the combination of objects, which is why it’s called the combination formula. The N factorial in the numerator is the number of ordering those N objects in the collection. But since we’re only selecting K of them, there are N minus K objects that we are not selecting, and there are N minus K factorial ways that those objects can be ordered. Since we’re not selecting them, we don’t care about their orderings. We divide that number out.
However, we also don’t care about the number of ways of ordering the objects we do select because we’re just interested in the combination, not the order. Since we’re selecting K objects, we’ve got to divide out that K factorial number of ways to arrange each combination because we only want to count each combination once. So that’s where it comes from. There are lots of interesting properties and uses for the combination formula or binomial coefficients, and we’ll certainly see more of those in the future.