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Permutation vs Combination: Is Order of Objects in a Set Important?

Mar 16,22

Permutation and Combination – both represent objects that are part of a set. The only difference is that in a permutation, we select objects or data of a group in a certain order while in a combination, the order of the objects or data does not matter. Both permutation and combination are important mathematical concepts and we will discuss them in detail here.

Permutation-Formula

Permutation Formulae

As we mentioned earlier, the order of things plays an important role in the permutation.

Its basic formula is:

P (n,r) = n!(n−r)!

where,

  • P is Permutation or Permutation Replacement (or the number of ways to choose a sample of r elements from a set of n distinct objects where order does matter. If replacements are allowed, it is called Permutation Replacement. If replacements are not allowed, it is only called Permutation).
  • n is the set or population
  • r is a subset of n or sample set

Derivation:

In a set of n different things, if you take out r things at a time, the number of Permutations you can get is P (n,r). It can also be written as nPr.

We can understand it like this:

Suppose there’s a large box with n different types of objects. You also have r number of empty boxes and you have to put one object in them from the box containing n objects.

You can pick any object to put in the first empty box. Hence, the number of ways you can fill it is: n

When you go to fill the second box, you will have one less object to choose from (as one object is already in the first box). So, the number of ways you can fill the second box is: n-1

Similarly, the number of ways you can fill the third box is: n-2

For the last box or rth number of box, the number of ways you can fill it is: (n-(r-1)), where r is the number of boxes.

When you add all these (the number of ways in which you can fill r boxes in succession), the number of permutations you get is:

n (n – 1) (n – 2) (n-3) . . . (n – (r – 1))

This can be also written as:

n (n – 1) (n – 2) … (n r + 1)

So, if you take r things at a time from the n number of different objects in a set, where 0 < r ≤ n (the number of objects taken in the sample is equal to or less than the number of objects in the set but is always more than zero), then the number of permutations you can get is:

⇒ nPr = n ( n – 1) ( n – 2)( n – 3). . .( n – r + 1)

This is what is written as:

P (n,r) = n!(n−r)!

Combination Formulae

In Combination, the order of things does not matter.

Its basic formula is:

C (n,r) = n!(r!(n−r)!)

where,

  • C is Combination or Combination Replacement (or the number of ways to choose a sample of r elements from a set of n distinct objects where order does not matter. If replacements are allowed, it is called Combination Replacement. If replacements are not allowed, it is only called Combination).
  • n is the set or population
  • r is a subset of n or sample set

Derivation:

In a set of n different things, if you take out r things at a time, the number of Combinations you can get is C (n,r). It can also be written as nCr.

We can understand it like this:

Suppose there’s a large box with n different types of objects. You also have r number of empty boxes and you have to put one object in them from the box containing n objects.

You can pick any object to put in the first empty box. Hence, the number of ways you can fill it is: n

When you go to fill the second box, you will have one less object to choose from (as one object is already in the first box). So, the number of ways you can fill the second box is: n-1

Similarly, the number of ways you can fill the third box is: n-2

For the last box or rth number of box, the number of ways you can fill it is: (n-(r-1)), where r is the number of boxes.

When you add all these (the number of ways in which you can fill r boxes in succession), the number of permutations you get is:

n (n – 1) (n – 2) (n-3) . . . (n – (r – 1))

This can be also written as:

n (n – 1) (n – 2) … (n r + 1)

Now, here’s where the difference between Permutation and Combination starts.

In Permutation, you pick objects in a certain order. In Combination, you can choose objects in any order. Hence, the number of ways in which each object in a sample can be picked is r! (where r is the number of objects in a sample.

Permutation vs Combination

Hence,

nCr = nPrr!

⇒ nCr = n!(n−r)!r!

This is what is written as:

C (n,r) = n!(r!(n−r)!)

Thus, we can also say:

nPr = nCr × r!

Permutation vs Combination: How different are they?

 

Criterion Permutation Combination
Meaning Permutations refer to different ways of arranging a set of objects in different sequential orders. Combinations refer to several ways of choosing items from a large set of objects. The order does not matter while choosing these items.
What is it? Ordered elements Unordered sets
Denotes Arrangement Selection
Order Relevant Irrelevant
Usual Application Permutations are often used for things of different kinds. Combinations are often used for things of a similar kind.
You can use it to answer How many different arrangements can be created from a given set of objects? How many different groups can be chosen from a larger group of objects?
Derivation A single combination can yield multiple permutations. A single permutation can only yield a single combination.
Example Permutation of three items a, b, c can be six in number if you choose two items in a sample at a time: ab, ba, ac, ca, bc, cb A Combination of three items a, b, c can be three in number if you choose two items in a sample at a time: ab, bc, ca

Applications of Permutation and Combination in Real Life

In simple words, Permutation is the arrangement of objects. Here’s an example:

You have to arrange five different books on a shelf. You can arrange them in different ways. You have five choices for the first book, four for the second book (since one is already sitting in the first position), three choices for the third book, and so on.

When you multiply 5*4*3*2*1, you get 120 different permutations in which these books can be arranged.

In simple words, Combination is the selection of objects. Here’s an example:

You work in a pizzeria. You offer customers:

  • 3 types of Cheese: American, Swiss, and Cheddar
  • 3 types of Crust: Thin, Regular, and Thick
  • 5 types of Toppings: Pepperoni, Sausage, Mushrooms, Cheese, and Onions

So, if a customer orders a cheese pizza, how many options can you offer?

3*3*5 = 54 combinations

It would not matter in what order customers choose their options. You combine them to get their pizza ready.

Do you know why the chances of winning a lottery are so slim? You will have to consider all the ways in which you can arrange the digits 0-9 in a given number of slots or their permutations!

We use Permutation and Combination regularly in many fields, such as:

Combinatorics

The field of Combinatorics is the mathematics of counting. In this field, combinations are used to determine the number of different ways that a set of objects can be arranged. It is used in search string algorithms and also natural language processing (NLP).

In search string algorithms, Combinatorics is used to search for a string within another string. For example, if someone was searching in the dark and only knew that they had seen something with three lights on it, then combinatorics can be used to determine how many different possibilities there are for where this item could be.

Combinations are also important when using natural language processing (NLP) to determine the probability of a word being used in a sentence. It is then possible to search for specific words or phrases that may be important within a larger piece of text.

Communication Networks

Permutations are used in communication networks to create unique identifiers for each packet of data. A combination is also used to create a password.

Many businesses use permutations and combinations in their passwords, often using a mix of letters and numbers. This makes it difficult for someone to hack into their account or steal their information.

Another way that permutations are used in a business is in the creation of product codes. Each product has a unique code that identifies it, and this code is created by a permutation.

This system ensures that no two products have the same code, and it also makes it easy to track products when they are being shipped or sold.

Combinations can be used in other areas of business as well. They are used to generate random numbers and in the creation of employee ID cards that have unique identifying codes for each person.

These types of permutations and combinations can also be found in robots or machines. Robotic arms use a combination code system, where a sequence is created randomly with different values for each joint.

This type of combination code ensures that every robotic arm is unique and cannot be duplicated by another robot or machine. It also makes it easy for robots to calculate movement, without requiring a computer chip in each one.

Computational and Molecular Biology

Permutations and combinations are also used in the study of genetics where chromosomes can be arranged into different patterns to determine their function or genetic makeup. This field of science uses permutations and combinations extensively to help understand how genes work within cells.

Genetic sequencing makes use of both permutation and combination as well to sequence the nucleotides within a DNA molecule. This process is important for understanding how different genes work and interact with each other.

One-dimensional sequencing in atoms, molecules, proteins, and other biomolecules uses permutations to represent the exact location of each atom.

Computer Architecture

Permutations and combinations are also used in the field of computer architecture. In this area, permutations are used to create unique addresses for each block of memory, while combinations are used to generate random passwords.

A computer architecture design is based on the arrangement of logic gates. Computer chips are designed based on desired outcomes for different permutations of input to output pins.

Computing Processes

Permutations are also used in computing processes, including data encryption. Data encryption allows people to send each other private messages or store sensitive information on their computers without worrying about anyone else being able to access it. This process makes use of a lock where the combination is based upon every letter, number, and symbol that make up an individual message or file.

This is important because it ensures that no one except for the sender and recipient of a message can read its contents or use any information contained within it. Data encryption allows people to communicate with each other more easily, securely, and efficiently than ever before by using permutations and combinations as an integral component of their process.

Cryptography

Cryptography is the practice of secure communication in the presence of third parties. It is used in a variety of different applications, including email, file sharing, and online banking.

One way that cryptography is used is through the use of permutations. In this application, a message is encrypted by mixing it with a random sequence of numbers or letters. This makes it difficult for someone to hack into the message and read it.

Another way that cryptography is used is through the use of combinations. As discussed above, mixing a sequence of letters and numbers makes it difficult for someone to guess the password and access the information that is stored in the account.

Cryptography is an important tool for keeping information safe and secure, and it is used in a variety of different applications. By using permutations and combinations, businesses can protect their data from being accessed by unauthorized individuals.

Databases and Data Mining

Databases and data mining also use permutations and combinations to organize and understand large sets of data. In a database, permutations can be used to create unique identifiers for each row or record within the database.

This is important because it makes it possible to track changes that may occur over time or between different versions of the same database. Data mining uses combinations to determine the probability of an event occurring within a set.

Homeland Security

After 9/11, the United States government created the Department of Homeland Security to better protect the nation from future terrorist attacks. This department uses a variety of methods to keep our country safe, including permutations and combinations.

One example is the use of facial recognition technology at airports. This technology compares each person’s face against a database of known terrorists or criminals to determine whether they pose a threat to the United States.

Permutations and combinations help make this process possible because it helps determine how effective these systems are at spotting potential threats quickly. The more accurate these computer programs can be, the safer everyone will be when traveling by air or through other parts of our country’s transportation system.

Network Security

Network security is one of the biggest concerns for businesses and organizations. One way that network security can be increased is by using permutations and combinations.

Using the password system is one way to protect a computer from unauthorized access to its files, but it only works if the password is difficult enough to guess or hack into. By creating a unique sequence of letters, numbers, and special characters with permutations and combinations, the password becomes much more difficult to crack.

This type of security system can be used for other types of accounts as well, making it possible to secure any file or piece of information that is stored on a computer network. It also makes it easier for employees within the company to access their files without having to worry about someone else getting into their account and stealing their information.

By using permutations and combinations, businesses can increase the security of their computer networks and protect their data from being accessed by unauthorized individuals. This is an important step in keeping company information safe and secure.

Operations Research

Operations research is the application of scientific methods and techniques to the planning, analysis, and improvement of business operations. This process uses a variety of mathematical models to help businesses make better decisions about how they can improve their processes.

One type of model that is often used in operations research is called a linear programming model. This model helps businesses determine the most efficient and cost-effective way to meet the demand for their products or services. Linear programming models use a variety of mathematical techniques.

Companies like IBM use permutations and combinations to help compute the potential success of new products or services. This is important because it helps companies decide whether an idea will be profitable before they invest too much time, money, or energy into developing it further.

This is also beneficial for consumers because businesses are more likely to succeed when they use data science to determine how promising a new product or service is. This means that people are more likely to see new and innovative products hit the market in the years to come, as businesses become better at using permutations and combinations to make sound decisions about their future.

These mathematical concepts also help with resource optimization. This process helps companies determine the best way to distribute limited resources (like materials, time, or money) to achieve their objectives. This can be applied to anything from allocating employees at a company to scheduling patients for surgery.

Scheduling jobs is a complex problem. It can be difficult to find the best way to complete all of the required tasks in the shortest amount of time while ensuring that each job is completed correctly. This is where permutations and combinations come into play. There are many different ways to schedule jobs, and trying to determine the best one can be overwhelming. However, by using permutations and combinations, businesses can quickly test different scenarios to see which one is most effective.

This allows them to find the best solution relatively quickly so that they can start saving time and money right away. By using these mathematical concepts, companies can improve their operations in several ways and become more efficient in the process.

Pattern Analysis

Permutation and combination are also important to pattern analysis. In this area, permutations can be used as a type of fingerprint that is unique for each item or piece of information.

Patterns in data make it possible to identify specific items within a larger set by creating specific patterns from the data points themselves. This makes it easier to find specific information that is needed.

Scientific Discoveries

Permutations and combinations have also been used to make scientific discoveries. In one well-known example, Rosalind Franklin used a combination of X-rays and math to determine the structure of DNA.

This discovery was later used by James Watson and Francis Crick to create their model for the double helix structure of DNA. Without the use of permutations and combinations, this discovery may never have been made.

Recently, the quick roll-out of COVID-19 vaccines would not have been possible without permutations and combinations to study their potential side effects. This discovery allows doctors and nurses all over the world to administer vaccines more quickly, safely, and effectively than ever before.

Simulation

Simulation models of different scenarios are another application of permutations and combinations. These models allow businesses to see how different decisions will impact their bottom line. This can be done for things like inventory management, financial planning, or customer service.

Simulations can also help companies predict the results of possible mergers and acquisitions. This is important because it allows businesses to make informed decisions about whether or not a merger is right for them.

Simulations can also help businesses optimize their supply chains. This means that businesses can get the most value out of the resources they have available to them by routing them in the most efficient way possible.

Besides businesses, simulation helps in making the world a safer place for all of us. Transportation safety, disaster relief plans, and public safety are just a few examples of areas that use simulation to save lives.

Traffic Lights

One more area where permutations are used in real life is in traffic lights, which rely on a combination of red, yellow, and green to let drivers know when they can go or stop.

A permutation is used in traffic lights that has three different colors: red, yellow, and green. The order of these colors determines whether a person should stop their car at the intersection or keep driving through it. In this example, six possible permutations can be used.

This system is very important for ensuring the safety of drivers and pedestrians, as it allows people to know what they need to do to avoid accidents.

So, if you are out to find a Math chapter that you are going to use in real life, Permutation and Combination is definitely the one that you will be using time and again.

 

March 16, 2022

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