Difference Between Permutation and CombinationIn the wide field of Mathematics, Permutation, and Combination are two basic concepts that show various disciplines of mathematics, from probability theory and statistics to combinatorics and algebra. Their combinatorial nature interconnects both these concepts (Permutation and Combination) and has different characteristics and applications. So, in this article, we will learn about the concept of Permutation and Combination, definitions, properties, applications, and differences, and end this article with a conclusion. Definition of PermutationA permutation is a mathematical concept that deals with the arrangement of objects in a special order; in other words, a Permutation of a set of objects refers to an arrangement of those objects in a distinct order. For example, consider a set of elements {A, B, C}. The permutations of this set would be ABC, ACB, BAC, BCA, CAB, and CBA. In Permutations, the order of arrangement is important; if we alter the order of the results, it is a different permutation. In terms of mathematics, if we have an 'n' number of different objects and we want to arrange them in 'r' positions, then the number of permutations is denoted by "^{n}P_{r }or n P r." It is calculated by the formula: n P r = n! / (n  r)!. Where 'n!' read as n factorial, is the product of all positive integers from 1 to 'n,' and (n  r)! represents the factorial of the difference between 'n and r.' This formula ensures that the arrangement is distinct (without repetition) and ordered. We can also write n! = n (n1) (n2) … 1 and (n  r)! = (n  r  1) (n  r  2) (n  r  3) … 1. Types of PermutationThere are two types of Permutation based on the arrival of objects, which are:
Let's discuss these types one by one. 1. Permutations with Repetition We can understand this with the help of an example. Consider we have six (6) objects, and we have to select these objects, so in "How ways can we select these six (6) objects (with repetition)?" We can solve this with 6^{6} = 6 x 6 x 6 x 6 x 6 x 6 = 46,656. So, we can select 6 objects in 46,656 ways with repetition. We can also understand this as 1^{st} object can be selected in 6 ways, 2^{nd} object can be selected in 6 ways, 3^{rd} object can be selected in 6 ways, 4^{th} object can be selected in 6 ways, 5^{th} object can be selected in 6 ways, 6^{th} object can be selected in 6 ways, this happened because repetition is allowed. So, we have this formula: 6! = 46,656 ways. So, the general formula becomes n^{n}. 2. Permutations without Repetition In this Permutation, we will use the general formula to solve these types of problems in which repetition is not allowed. We will use the above example to put this in use, as we had six objects, and we had to select these objects one by one (but without repetition). So, we will put n = 6 and r = 6 in the abovedescribed formula; we have 6 P 6 = 6! / (6  6)! = 6! = 720 ways. Note: We have to remember that 0! = 1.Permutations find many applications, such as solving probability problems involving arrangements, cryptography, organizing events, and generating passwords. For example, when selecting a president, vice president, and secretary from 10 candidates, the number of permutations can be calculated to determine the possible officer arrangements. Definition of CombinationCombination, on the other hand, deals with the selection of objects from a set without considering the order of arrangement; in contrast to permutations, where ABC is distinct from BAC, these arrangements are similar in the concept of combinations since they contain the same elements. A combination represents the selection of 'r' objects from a set of 'n' without regard to the order of selection. Mathematically, the number of combinations denoted as "n C r or ^{n}C_{r}" is given by the formula: n C r = n! / r! ⋅ (nr)! The denominator consists of the product of 'r!' and '(n  r)!', showing that the overcounting of equivalent combinations is not included. Combinations find applications in probability calculations involving the selection of items and in various fields like genetics, where the combination of alleles leads to genetic diversity. For instance, when forming a committee by selecting 3 members from a group of 10, you would use combinations to determine the number of different committee compositions. Key DifferencesWe will discuss the differences between Permutations and Combinations in this paragraph:
When to Use Which
Relation Between Permutation and CombinationWe will use formulas of Permutation and Combination and establish a relation between these two concepts. As, n P r = n! / (n  r)! Multiply and divide by 'r!' in the numerator and denominator on the righthand side. We get n P r = r!. n! / r!. (n  r)! As we know, n C r = n! / r!. (n  r)! This implies n P r = (r!). n C r So, we have a proper relationship between Permutation and Combination. Applications of Permutation and CombinationsThere are so many applications of these concepts in various fields other than mathematics; some of them are discussed below description:
ConclusionPermutations and combinations are interconnected by their combinatorial nature, and this serves distinct purposes in the field of mathematics and its applications in various fields. Permutations deal with the arrangement of objects in a specific order, while combinations focus on the selection of objects without consideration for order. These concepts' formulas, properties, and applications are different, making them invaluable tools for solving various problems across various disciplines. Understanding the difference between permutations and combinations helps solve mathematical and scientific reallife problems.
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