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Introduction
Different ways of arranging objects in a specific order are known as permutations. It can alternatively be defined as the reordering of items in a previously ordered set in a linear order. The symbol represents the number of permutations of n different items taken r at a time. It encrypts bus, train, and aircraft timetables, as well as zip code and phone number assignments. Permutations are employed in a few different contexts. The word permutation is derived from the term “permute” which means to change the order of something. Let’s have a tour to the world of permutation.
What is permutation?
When the order of the arrangements counts, a permutation is a mathematical technique for determining the number of alternative arrangements in a collection. Choosing only a few items from a group of items in a specific sequence is a common mathematical problem. For example, there are five chairs, for example, and three people must be seated. There are five different methods to seat the first person, four different ways to sit the second person, and three different ways to seat the third person. We multiply the alternatives accessible to us to get the number of ways to arrange 3 people on 5 chairs. We do it in five different ways: 5*4*3. It can be done in 60 different ways, in other words. It can be written as- (5!) / (2!) (or) (5!) / (5 – 3)!
We get n alternatives to fill the first chair, n-1 options to fill the second, and n-2 options to fill the third chair if we generalise this. As a result, the total number of permutations (arrangements) of r individuals in n chairs is- nPr = n! / (n – r)!
Formulas of permutation
In the last section, we looked at the basic permutations formula. The following are many permutation formulas that are utilized in various contexts.
- The formula gives the number of permutations (arrangements) of ‘n’ different objects, where ‘r’ things are taken at a time and repetition is not allowed: nPr = n! / (n – r)!.
- According to the circular permutation formula, there are (n-1) ways to arrange ‘n’ items in a circular shape.
- By using the permutation formula, the total number of ways in which n different things can be arranged (taking all at a time) is n! (this is because nPn = n! / (n – n)! = n!/0! = n!/1 = n!).
- Using the above formula, the total number of ways of arranging n different things (taking all at a time) is n! (this is because nPn= n! / (n – n)! = n!/0! = n!/1 = n!).
Example: What are the different ways can a president, a treasurer and a secretary be chosen from among 7 candidates?
There are 210 methods to select a president, treasurer, and secretary from a pool of seven candidates.
Example: (a) How many words can you make out of the letters in the word TRIANGLES? (a) How many of these words begin with the letter T and end with the letter S?
Solution: a. There are 9 distinct letters in the given word. Thus, the number of different permutations (or arrangements) of the letters of this word is 9P9 = 9!.
- If we fix T at the start and S at the end of the word, we have to permute 7 distinct letters in 7 places. This can be done in 7P7= 7! ways. Thus, the number of such words is 7!
Example: Find the number of words, with or without meaning, that can be formed with the letters of the word ‘JAPAN’.
Solution : Total words in letter ‘JAPAN’ =5
Repeated letters (A) – 2
We divide the factorial of the number of all letters in the word by the number of occurrences of each letter when a letter appears more than once in a word.
Therefore, the number of words formed by ‘JAPAN’ = 5!/2! = 60.
Example: How many different words can be formed with the letters of the word ‘SUPER’ such that the vowels always come together?
Solution: Total number of words in word ‘SUPER’ is 5
To determine the number of permutations that can be created when the vowels U and E are combined.In some circumstances, we combine the letters that should be grouped together and treat them as a single letter.
S,P,R (UE). The total word=4.
As a result, there are 4 different methods to arrange four letters!
There are two different ways to order U and E in U and E!
As a result, there are 4! * 2! = 48 different ways to arrange the letters of the ‘SUPER’ so that vowels are always together.
Questions for practice
Q1. Find the number of permutations of the letters of the word ‘CONSTANT’ such that the vowels always occur in odd places.
Q2. If all permutations of the letters of the word REGAIN are arranged in the order as in a dictionary. What is the 49th word?
Q3. 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?
Q4. A student has to answer 10 questions, choosing at least 4 from each of Parts A and B. If there are 6 questions in Part A and 7 in Part B, in how many ways can the student choose 10 questions?
Q5. In a movie theatre, three married couples will be seated in a row with six seats. How many different ways may husband and wife be seated next to one other? If all the ladies sit together, count the number of ways they can sit.
Conclusion
Permutations are pre-determined combinations of things that can be repeated or not. They’re calculated using the formula nPr = n! / (n – r)!, where r is the number of things taken at once. Permutations are considered when things are to be sorted, ordered, or positioned.
