Permutation

A permutation refers to the arrangement of objects in a specific order. The order of arrangement is important in permutation. A permutation let us know how many different ways a set or number of things can be arranged.

For example, if we have three different numbers 1, 2, and 3. The permutations of these three numbers are: , and .

In general, the number of permutations of ‘n’ items can be calculated using the factorial function, denoted by , which is the product of all positive integers less than or equal to . So, if you have different items, there are ‘n’ choices for the first item, choices for the second item, choices for the third item, and so on, until there is only one choice left for the last item.

Here are a few practical examples of permutation along with their applications:

1. Password Creation In computer science, permutation is frequently used in creating unique passwords or codes. For instance, if you have to create a 4-digit PIN, the number of permutations possible using the digits 0-9 is . This is because there are 10 options (digits 0-9) for each of the 4 places in the PIN.
2. Sports Events In sports, permutation is often used to determine the number of possible outcomes in a tournament. For instance, in a football league with 20 teams, the number of different ways the teams can finish the season (from 1st to 20th place) is a permutation of 20, which equals approximately .
3. Travel Planning When planning a trip with multiple destinations, permutations can help determine the total number of possible routes. For example, if you are planning a tour to 5 cities and want to know how many different routes you can take, the total number of permutations is (5 factorial), which equals 120.
4. DNA Sequencing In biology, permutation is used in DNA sequencing. For instance, a strand of DNA contains 4 types of nucleotides. The number of different sequences in which these nucleotides can appear is a permutation of 4, which equals for a sequence of 4 nucleotides.
5. Arranging Books on a Shelf If you have 7 books and you want to arrange them on a shelf, the number of different arrangements possible is 7!, which equals 5040.

Quizzes

In a social network analysis, a researcher wants to analyze the order in which a message travels between 5 distinct people (A, B, C, D, and E). How many possible permutations are there for the order of message transmission?
a) 120
b) 60
c) 24
d) 12

Consider a scenario where a committee of 4 members needs to choose a president, a vice-president, a secretary, and a treasurer. How many different ways can the committee assign these positions?
a) 16
b) 24
c) 256
d) 4

An engineering team is designing a control panel with 7 different switches. In how many different ways can the switches be arranged on the panel?
a) 5,040
b) 2,520
c) 40,320
d) 10,080

A factory has 6 different machines and they need to be arranged in a specific sequence for an optimal workflow. How many possible arrangements are there?
a) 720
b) 360
c) 120
d) 60

A biologist is studying the possible arrangements of a specific protein made up of 5 distinct amino acids. How many unique permutations of these amino acids are possible?
a) 25
b) 100
c) 120
d) 150