Mathematical Preliminaries: Arrangement and Selection Problems

Basics of Arrangement

Arrangement problems primarily deal with combinations and permutations of sets or sequences. They may also be referred to as combinatorial problems.
A permutation of a set is any arrangement of its elements. The order of the elements in a permutation is important.
The number of permutations of a set of n elements is n! (n factorial), which is the product of all positive integers less than or equal to n.
A combination of a set is an selection of its elements. Unlike permutations, the order does not matter in a combination.
The number of combinations of r items from a set of n elements is given by nCr = n! / [(n - r)! * r!]

The factorial of a non-negative integer n (denoted as n!), is the product of all positive integers less than or equal to n.
It acts as an essential tool while dealing with permutations and combinations.
Factorial of zero is defined as one i.e., 0! = 1

A permutation is an arrangement from a set where the order is important.
The number of permutations of n things taken r at a time is denoted as nPr = n! / (n - r)!
A combination is a selection from a set where the order is unimportant.
The number of combinations of n things taken r at a time is denoted as nCr = n! / [(n - r)! * r!]

Arrangements of distinguishable objects are governed by the rules of permutations and combinations, focusing on the placing or selecting of unique, distinguishable items.
Arrangements of indistinguishable objects, such as anagrams, use multiset coefficients to account for indistinguishability.
When arranging n indistinguishable objects into r distinct boxes, use a stars and bars argument to consider possible divisions.

Circular permutations refer to arrangements around a circle, where there is no ‘start’ and ‘end’ to the set. There are (n-1)! ways to arrange n objects in a circle.
Permutations with repetition are used when some items within the set are the same (indistinguishable). Divide the total permutation count (n!) by the factorial of the count of each repeated element.
The Binomial theorem describes the algebraic expansion of powers of a binomial and can also be seen as a combination problem where it can be applied.