the arrangement of different objects into a linear order using each object exactly once is called a permutation of these objects. the number of all permutations of objects is called factorial, and is denoted by .
in a permutation, order matters, repetition not allowed.
let be a permutation, and let . then there exists a positive integer so that .
the number of permutations for a given set containing elements is
assuming there are 3 girls and 4 boys, we want them seated on a bench
such that the 3 girls are next to each other
if it wasnt for the condition that the girls must be seated next to each other we would have permutations since the girls are seated next to each other, its safe to look at the girls as a single element in a set of 5 such that each boy takes a seat and the fifth seat is reserved for a single girl, which means we have permutations for this set now since the girls can switch places and it wouldnt break the condition, because they'd still be seated one next to the other, this property has permutations, so the total number of permutations for the seat is
such that the 4 boys sit all next to one another
the permutations of the boys sitting next to each other is , because in total we'd have 3 girls and 1 boy (instead of 4), and the total permutations is so the answer is
permutation with repetition
given items and types, of the first type, of the second type and so on.. such that , then the total number of possible permutations with items is
circular permutation
when the permutations are circular, the number of options becomes