In combinatorial mathematics, a<unk>  (more fully, a balanced incomplete<unk>  is a particular kind of set system, which has long-standing applications to experimental<unk>  (an area of statistics) as well as purely combinatorial aspects.
Given a finite set X (of elements called points) and integers k, r, λ ≥ 1, we define a 2-design B to be a set of k-element subsets of X, called<unk> s, such that the number r of<unk> s containing x in X is independent of x, and the number λ of<unk> s containing given distinct points x and y in X is also independent of the choices.
Here v (the number of elements of X, called points), b (the number of<unk> s), k, r, and λ are the parameters of the<unk> .  (Also, B may not consist of all k-element subsets of X; that is the meaning of incomplete.)  The<unk>  is called a (v, k, λ)-design or a (v, b, r, k, λ)-design.  The parameters are not all independent; v, k, and λ determine b and r, and not all combinations of v, k, and λ are possible.  The two basic equations connecting these parameters are
A fundamental theorem (Fisher's inequality) is that b ≥ v in any<unk> .  The case of equality is called a symmetric<unk> ; it has many special features.
Examples of<unk> s include the lines in finite projective planes (where X is the set of points of the plane and λ = 1), and Steiner triple systems (k = 3).  The former is a relatively simple example of a symmetric<unk> .
Block design
Design of experiments
Evidence-based design
Agent study
Cumulative incidence
Surrogate endpoint
Action study
Stratified sampling