Uses of Permutation and Combination

One of the most significant ideas in mathematics is permutation and combination, which has applications in science, engineering, and research. 

The distinction between permutation and combination is that permutation is when a collection of data is chosen from a specific group, whereas combination is the order in which the data is arranged. As a result, we can define permutation as an ordered combination.

Permutation and combination are tools that can be used to group data and understand it. Students studying mathematics should be well-versed in the principles because they are used in every advanced research and data analysis.

This issue is also covered extensively in a variety of national and state-level competitive exams. This article discusses permutation and combination issues, as well as solutions, permutation and combination formulas, and more. To gain a thorough understanding of the principles, read the entire essay.

Definition of Permutation and Combination

The mathematical definitions of permutation and combination are as follows:

What is permutation, exactly?

A permutation is a set of objects taken in a specific order, some or all at once. With permutations, every little detail counts. It implies that the order in which elements are placed matters.

Permutations are divided into two categories:

1. Repetition is permitted: in the case of the number lock, it may be “2-2-2.”

2. Repetition is not permitted: The first three people in a race, for example. It is not possible to be first and second at the same time.

What is the definition of combination?

Combination is a method of selecting components from a set without regard for their sequence of selection. Only the elements chosen matter in the combination. It implies that the sequence in which elements are chosen isn’t critical.

Combinations can be divided into two categories:

1. Repetition is permitted, such as with coins in your pocket (2,5,5,10,10)

2. Repetition is not permitted: Lottery numbers, for example (2,14,18,25,30,38).

Formula for Permutation and Combination

Permutation and combination problems are solved using a variety of formulas. Here is the complete list of permutation and combination formulas:

Permutation Formula Derivation

Let’s pretend there are r boxes, each of which can hold one item. There will be as many combinations as there are methods to fill r empty boxes with n things.

– Number of options to fill the first box: n

– Number of possibilities to fill the second box: (n – 1)

– Number of possibilities to fill the third box: (n – 2)

– Number of possibilities to fill the fourth box: (n – 3)

– Number of possibilities to fill the rth box: [n – (r – 1)]

n(n – 1)(n – 2)(n – 3)… (n – r + 1) is the number of permutations of n different objects taken r at a time, where 0 r n and the objects do not repeat.

nPr = n(n – 1)(n – 2)(n – 3).

We get: by multiplying and dividing by (n – r) (n – r – 1)… 3 ×2 ×1

Formulas for Combinations

When repetition isn’t an option: C is a collection of n different things done one at a time (order is not important). C is defined as

When repetition is permissible: C is a collection of n different things done r at a time with repetition (order is irrelevant). C is defined as:

Combination Formula Derivation

Let’s pretend there are r boxes, each of which can hold one item.

– Number of options for selecting the first object from a set of n objects: n

– Number of methods to choose the second object from among (n-1) different options: (n-1)

– Number of methods to choose the third object from (n-2) different options: (n-2)

– The number of methods to choose the rth object among [n-(r-1)] different objects: [n-(r-1)]

An ordered subset of r elements is created by completing the selection of r things from the initial set of n things.

∴ There are n (n – 1) (n – 2) (n-3)… (n – (r – 1) ways to select r elements from the original set of n elements. (n – r + 1).

Consider the ordered subset of r elements, including all permutations. This subset’s total number of permutations equals r! because every combination of r objects can be rearranged in r! different ways

As a result, (nCrr!) is the total number of permutations of n different items taken r at a time. It’s all about nPr.

Conclusion

Combination is a method of selecting components from a set without regard for their sequence of selection. Only the elements chosen matter in the combination. It implies that the sequence in which elements are chosen isn’t critical.

The distinction between permutation and combination is that permutation is when a collection of data is chosen from a specific group, whereas combination is the order in which the data is arranged. As a result, we can define permutation as an ordered combination.