Permutations refer to the arrangement of objects in a specific order, while combinations refer to the selection of objects without considering the order. In other words, permutations involve rearranging all the elements, while combinations involve selecting some elements from a larger set.
Permutations are used when the order of the elements matters, such as in finding the number of ways to arrange letters in a word. Combinations are used when the order of the elements does not matter, such as in selecting a group of people for a committee.
The formula for calculating permutations is n!/(n-r)! where n represents the total number of objects and r represents the number of objects being arranged. For example, if you have 8 letters and need to arrange 3 of them, the calculation would be 8!/(8-3)! = 8!/5! = 8*7*6 = 336 permutations.
The formula for calculating combinations is n!/r!(n-r)! where n represents the total number of objects and r represents the number of objects being selected. For example, if you have 10 students and need to select a group of 4 for a project, the calculation would be 10!/(4!*(10-4)!) = 10!/(4!*6!) = 210 combinations.
Permutations and combinations are used in many fields, including mathematics, statistics, and computer science. In real-life, they are used in various scenarios such as lottery number combinations, password combinations, and sports tournament brackets. They are also used in analyzing data and making predictions in fields such as genetics and finance.