Permutation as an Arrangement

Permutation

A permutation of a set is a loosely defined organization of its members into a sequence or linear order, or if the set is already ordered, a rearranging of its elements in mathematics. The act of shifting the linear order of an ordered set is also referred to as “permutation.”

Permutations are distinct from combinations, which are random picks of members from a set. There are six permutations of the set 1, 2, 3 written as tuples, for example: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 1, 2). (3, 2, 1). This three-element set has every possible ordering. Anagrams of words with different letters are also permutations: the letters in the original word are already ordered, and the anagram is simply rearranging them. In the domains of combinatorics and group theory, the study of permutations of finite sets is a crucial topic.

Permutations are employed in practically every branch of mathematics as well as a wide range of other scientific disciplines. They are used in computer science to analyze sorting algorithms, in quantum physics to describe particle states, and in biology to describe RNA sequences.

Permutation Formula

The following is the permutation formula of n objects for r selection of objects:

nPr=n!(n-r)!

Combination

A combination in mathematics is a selection of elements from a set with distinct members, where the order of selection is irrelevant (unlike permutations). Given three fruits, such as an apple, an orange, and a pear, there are three possible combinations of two: an apple and a pear, an apple and an orange, and a pear and an orange. A k-combination of a set S is a subset of k different items of S, to put it another way. As a result, two combinations are identical if and only if they have the same members.

Combination Formula

The number of k-combinations in a set of n elements is

nCk=n!k! (n-k)!

Relation between Permutation and Combination

nCk=nPrk!

Uses of Permutation and Combination

• When an order/sequence of arranging is required, permutations are utilized. When simply the number of feasible groups needs to be identified and the order/sequence of arrangements isn’t important, combinations are employed.
• Permutations are used for a variety of things. Combinations are used to describe items that are similar in nature.
• pq, qr, rs, pr, qs, ps, qp, rp, sr, rp, sq, sp is the permutation of two items from four given objects p, q, r, s. pq, qr, rs, pr, qs, ps is a combination of two things from three given things p, q, r, s.

Some facts and results of permutation and combination

• When repetition of objects is allowed, the number of permutations of n things taken all at once is nn. When repetition of objects is allowed, the number of permutations of n objects, taken r at a time, is nr.
• The number of permutations of n objects in which p1 is of one kind, p2 is of a different kind,…, pk is of a different kind, and the rest, if any, is of various kinds is

n!p1!p2!……pk!

• Let n and r be positive integers with the property that r≤n. Then

nCr=nCn-r

nCr+nCr-1=n+1Cr

nn-1Cr-1=(n-r+1)nCr-1

Examples on how to use Permutation and Combination

1. There are 27 males and 14 girls in each class. For a function, the teacher wants to choose one male and one girl to represent the class. How many different ways can the teacher make this decision?

 Ans : The teacher must do two operations in this situation:

Choosing a boy from a pool of 27 boys and a girl from a pool of 14 girls.

The first of them can be done in 27 different ways, while the second can be done in 14 different ways. The required number of ways is 27✕14 = 378, according to the basic counting principle.

  2. Between 99 and 1000, how many numbers have 7 in the unit’s place?

 Ans : To begin, keep in mind that all of these numbers have three digits. The number 7 has taken the place of the unit. Any of the ten digits from 0 to 9 can be used as the middle digit. Any of the nine digits from 1 to 9 can be used in the hundredth place. As a result, there are 10✕9 = 90 numbers between 99 and 1000 that have 7 in the unit’s location, according to the fundamental concept of counting.

  3. How many numbers between 99 and 1000 have at least one digit that is seven?

 Ans: Total number of three-digit numbers with at least one digit of 7 = (Total number of three-digit numbers) − (Total number of three-digit numbers) (The total number of three-digit integers in which the number seven does not appear at all.)

=(9✕10✕10)-(8✕9✕9)

=900-648=252

  4. How many different ways can five children be organized in a line so that I two specific children are always together and (ii) two specific children are never together?

 Ans : We evaluate the arrangements by considering two specific children as one, and so the remaining four can be organized in 4! = 24 different ways. Again, two specific children can be grouped in two different ways. As a result, there are a total of 24✕2 = 48 possible arrangements.

There are 48 permutations of 5 children among the 5! = 120 possible combinations. Two specific children are never paired in the remaining 120 – 48 = 72 possibilities.

  5. If all permutations of the letters in the word AGAIN are organized in a dictionary-like manner. What is the 49th word in the sentence?

 Ans : There are 4! = 24 words starting with the letter A and arranging the other four letters. These are the first 24 words in the sentence. Then, starting with G, arrange A, A, I, and N in various ways to arrive at 4!2!1!1!=12 words. The 37th word begins with the letter I.

There are 12 words that begin with the letter I once more. This is true until the 48th word. NAAGI is the 49th word.

  6. How many different ways can three mathematics books, four history books, three chemistry books, and two biology books be stacked on a shelf so that all books on the same subject are together?

 Ans : First, we group together books on a single topic. As a result, there are 4 units that can be organized in 4! = 24 different ways. Mathematics books can now be arranged in 3! ways, history books in 4! ways, chemistry books in 3! ways, and biology books in 2! ways in each of the arrangements. As a result, the total number of ways is 4!✕3!✕4!✕3!✕2! = 41472.

Conclusion

In this article, you learned about Permutation and Combination, Permutation formula, Combination formula, and Uses of permutation and combination.