Permutations and combinations are both ways of counting the number of possible arrangements of a given set of objects. However, permutations take into account the order of the objects, while combinations do not. In other words, permutations are arrangements where the order matters, while combinations are arrangements where the order does not matter.
To calculate the number of permutations, you can use the formula n! / (n - r)!, where n is the total number of objects and r is the number of objects being selected for the permutation. For example, if you have 5 objects and want to find the number of permutations of 3 objects, the calculation would be 5! / (5-3)! = 5! / 2! = 5 * 4 * 3 = 60 permutations.
Permutations with repetition allow for the same object to be selected more than once in a permutation, while permutations without repetition do not. For example, when finding the number of ways to arrange the letters "ABC" with repetition, you can have "ABC", "ACB", "BAC", "BCA", "CAB", "CBA" as well as "AAB", "ACA", "BBC", etc. But in permutations without repetition, you would only have "ABC", "ACB", "BAC", "BCA", "CAB", "CBA".
Combinations differ from permutations in that they do not take into account the order of the objects. This means that in combinations, different arrangements of the same objects are considered to be the same combination. For example, the combination of choosing two letters from the set "A", "B", and "C" would include "AB" and "BA" as the same combination, while in permutations they would be considered two different arrangements.
Permutations and combinations have many real-life applications, such as in probability and statistics, computer science, and genetics. They can be used to calculate the number of possible outcomes in a game of chance, to generate unique passwords, and to analyze genetic variations and mutations. They are also commonly used in problem-solving and decision-making in various fields such as finance, marketing, and engineering.