Permutation and Combination

What are Permutation and Combination?

Permutation and combination are the different ways to represent a gaggle of objects by picking them during a set and forming subsets. It defines the different types of ways to set out a particular group of knowledge. Once when we select the info or objects from a particular group, it is said to be permutations, whereas the order during which they are represented is called a combination. Both concepts are very vital in Mathematics. 

By keeping in mind the ratio of the number of desirable subsets to the amount of all possible subsets for several games of chance during the 17th century, the famous French mathematicians Blaise Pascal and Pierre de Fermat gave impetus to the evolution of combinatorics and probability theories. 

The concepts and differences between permutations and combinations can be demonstrated by an examination of every different way in which a pair of objects can be chosen from five separable objects—like the letters A, B, C, D, and E.


What is Permutation?

In Mathematics, permutation helps to connect to the act of setting out all the members of a set into some kind of arrangement or structure. In other words, if the set is already arranged, then the rearranging of its elements is called the procedure of permuting. Permutations take place, in more or less eminent ways, in nearly all areas of Mathematics. They frequently arise when different types of arrangements on certain finite sets are considered.


What is a Combination?

The combination is a way of choosing objects from an assemblage such that (unlike permutations) the arrangement of choosing does not matter. In smaller cases, it is impossible to count the number of combinations. Combination refers to the amalgamation of n things taken k at a time without recurrence. To ask for combinations during which recurrence is permitted, the terms k-selection or k-combination with recurrence are frequently used.


Formulas

There are various formulas involved in the concept of permutation and combination. The 2 key formulas are:

Permutation Formula

A permutation is the selection of r things from a set of n things. It is chosen without replacement and where the order matters.

nPr = \[\frac{(n)!}{(n-r)!}\]

Combination Formula

A combination is the selection of r things from a set of n things. It is chosen without replacement and where the order doesn't matter.

nCr = \[\frac{(n)!}{r!(n-r)!}\]


Solved Examples

1. Find the number of permutations and combinations of n = 12 and r = 2?

Solution: 

Given,

n = 12

r = 2

By using the formula given above:

Permutation:

nPr = \[\frac{(n)!}{(n-r)!}\]

=  \[\frac{(12)!}{(12-2)!}\]

= \[\frac{12!}{10!}\]

=  \[\frac{(12×11×10!)}{10!}\]

= 132

Combination:

nCr = \[\frac{(n)!}{r!(n-r)!}\]

 = \[\frac{(12)!}{2!(12-2)!}\]

 = \[\frac{12!}{2!×10!}\]

=  \[\frac{12×11×10!}{2!×10!}\]

= 66

2. In how many ways does a committee that consists of 5 men and 3 women, can be chosen from 9 men and 12 women?

Solution:

Choosing 5 men out of 9 men = 9C5 ways = 126 ways

Choosing 3 women out of 12 women = 12C3 ways = 220 ways

The committee can be chosen in 27720 ways.


Did you know?

There are many practical real-life implements of permutation and combination. Here is one such real-life example:

Prior applications, it is important to mention that they are interesting! Like sudoku (or maybe a killer samurai sudoku), it can just be interesting. It can also enhance your thinking abilities.

Well, we can say it is used very excessively in reality. However, the theory likely isn’t used so much, but from a realistic point of view, when someone is hungry and has to make themselves a meal, from all the ingredients present in their home, there are a lot of combinations and permutations to observe.

When a person was preparing a sandwich, he required 2 slices of bread. There were 10 left. He could have paired those slices of bread in 10C2 (45) ways. And, if theoretically seen, each slice could have been on either top or bottom of the sandwich, so it could have been prepared in 10P2 (90) ways.

FAQs (Frequently Asked Questions)

1. How do you find good combinations and permutations? Give examples of permutations and combinations.

Ans. You need to revise and solve as many problems as you can each day to be able to understand the concept clearly. That is just the bare minimum. Everything is interconnected, so it is curious enough to dig deep into every question and problem every day. If there's one way of mastering it, it has to be the fact that one must not just solve questions but create them too. Keep asking/creating questions and work smart and hard in achieving the answers yourself. 

  • An example of permutations is the number of 2 letter words which can be formed by using the letters in a word say, GREAT;

5P2 = (n!)/(n-r)! = (5!)/(5-2)!

  • An example of combinations is in how many combinations we can write the words using the vowels of the word GREAT;

5C2 = (n!)/r!(n-r)! = (5!)/2!(5-2)!

2. What is the difference between Combination and Permutations?

Ans. A combination is a mathematical method that decides the number of achievable positioning in a group of objects where the sequence of the preference does not matter. In combinations, you might pick the items in any sequence. Combinations can be jumbled with permutations so one has to be very careful about it and make sure that the two different terms "combinations" and "Permutations" do not get jumbled. The idea of each term should be crystal clear to not make mistakes. 

To examine whether a question is a permutation or combination question, question yourself if the sequence matters. If the sequence of things is vital, then it is a permutation question, if the sequence doesn't matter then it is a combination question. Although no matter how easy it seems to differentiate between the two, it does become tricky at times to know if the question given is permutation or combination. Regular practice is the key to avoid such problems and uncertainty.