PERMUTATION AND COMBINATION

Permutations and combinations offer methods to represent a specific set of objects by selecting them within a group and forming subsets.

In mathematics, permutations and combinations offer various approaches to organizing a given set of data. The fundamental distinction lies in their definitions: a combination is the selection of objects from a group without considering their order, while permutations involve the consideration of the order of selected data.

This article delves into permutation and combination formulas, providing solved examples, definitions, and a comparison table of permutations versus combinations.

An additional crucial aspect is the repetition of data, leading to the classification of both permutations and combinations into categories of repetition and non-repetition. The article comprehensively explores these concepts within the realm of mathematics

What is Permutation?

In mathematics, a permutation refers to an arrangement or ordering of a set of distinct elements. It involves rearranging the elements in different orders, and the number of possible permutations depends on the size of the set. The order of arrangement matters in permutations, meaning that changing the order of the elements results in a different permutation.

The formula for the number of permutations of $n$ distinct elements taken $r$ at a time is given by:

where $n!$ (read as "n factorial") represents the product of all positive integers up to $n$. Permutations are often denoted as nP_r​ or $nPr$, representing the number of ways to arrange $r$ elements out of a total of $n$ distinct elements.

What is a Combination?

Combination refers to a selection of items from a larger group without regard to the order in which they are arranged. Unlike permutations, the order of selection does not matter in combinations. In other words, different arrangements of the same elements are considered equivalent in combinations.

The formula for the number of combinations of $n$ distinct elements taken $r$ at a time is given by:

where $n!$ (read as "n factorial") represents the product of all positive integers up to $n$, and $r!$ represents the factorial of $r$. Combinations are often denoted as nC_r​, $nCr$

Permutation and Combination Formulas

The formula for permutations is:

• nPr = n!/(n-r)!

where:

• n is the total number of elements.
• r is the number of elements you are choosing.
• ! represents the factorial symbol, meaning the product of all positive integers from 1 to the given number.

This formula basically tells you how many ordered arrangements you can make by choosing r elements out of n without repetition.

Combination:

The formula for combinations is:

• nCr = n!/[r! (n-r)!]

where the symbols have the same meaning as in the permutation formula.

This formula tells you how many unordered groups you can make by choosing r elements out of n without repetition. The order in which you choose the elements doesn't matter in a combination.

Here are some key differences between the formulas:

• The permutation formula has 1/(n-r)! in the denominator, which accounts for the overcounting of arrangements due to order not mattering.
• The combination formula also has (n-r)! in the denominator, which further corrects for the overcounting that occurs when choosing the same group of elements in different orders.

Examples:

• How many permutations are there of the letters ABC?
  • n = 3, r = 3
  • 3P3 = 3!/(3-3)! = 6 (ABC, ACB, BAC, BCA, CBA, CAB)
• How many combinations are there of 3 fruits from a basket of 5?
  • n = 5, r = 3
  • 5C3 = 5!/[3! (5-3)!] = 10 (apple, banana, orange; apple, banana, grapes; etc.)

 

Difference Between Permutation and Combination

 

Subject	Permutation	Combination
Order Matters	Yes, the order of arrangement matters.	No, the order of selection is irrelevant.
Formula	P(n,r)=n!/(n−r)!​	C(n,r)=n!/r!⋅(n−r)!​
Denotation	nPr​, $nPr$	nCr​, $nCr$
Example	Arranging books on a shelf.	Selecting a committee from a group.
Use Case	When the arrangement or order matters, e.g., seating arrangements.	When the arrangement or order is irrelevant, e.g., selecting a team.
Order Dependency	Different orderings are considered distinct permutations.	Different orderings are considered equivalent combinations.
Calculation	Involves dividing the factorial of the total elements by the factorial of the difference between the total elements and the chosen elements.	Involves dividing the factorial of the total elements by the product of factorials of the chosen elements and the difference between the total elements and the chosen elements.
Notation Example	$P(5,2)$ represents permutations of 5 elements taken 2 at a time.	$C(5,2)$ represents combinations of 5 elements taken 2 at a time.

 

 

 

 

Solved Examples of Permutation and Combinations   1.Permutations Example: Suppose we have 5 different books (A, B, C, D, E), and we want to arrange 3 of them on a shelf. How many different arrangements are possible? Solution: So, there are 60 different ways to arrange 3 books out of 5 on the shelf.   2.Combinations Example: Suppose we have 8 students, and we want to form a study group of 4 students. How many different study groups can be formed?   Solution: So, there are 70 different ways to form a study group of 4 students out of 8.