Combination

Combination is a term used across mathematics, science, and law to denote a selection of items from a larger set without regard to the order of selection, or a joining of separate elements into a single entity.

Mathematics

In mathematics, a combination is a selection of \( r \) elements from a set of \( n \) distinct elements, where the order of selection does not matter. The number of such combinations is given by the binomial coefficient:

\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]

This formula counts distinct groupings, often referred to as “\( n \) choose \( r \).” For example, the combinations of 3 elements taken 2 at a time from the set {A, B, C} are {A,B}, {A,C}, and {B,C} — three possibilities. Combinations are fundamental in probability, combinatorics, and algebra, and they contrast with permutations, where order matters.

Properties

Key properties of combinations include symmetry (\(\binom{n}{r} = \binom{n}{n-r}\)), Pascal’s rule, and the recurrence relation used to build Pascal’s triangle. Combinations also appear in the binomial theorem:

\[
(x + y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r
\]

Generalizations

The concept extends to combinations with repetition (multisets), counted by the formula \(\binom{n+r-1}{r}\). Other generalizations include multinomial coefficients and combinations in the context of graph theory and design theory.

Chemistry

In chemistry, a combination refers to a chemical reaction where two or more substances combine to form a single product, often called a synthesis or combination reaction. For example:

\[
2\text{H}_2 + \text{O}_2 \rightarrow 2\text{H}_2\text{O}
\]

In such reactions, the reactants may be elements or compounds, and the product has a different chemical composition. The term also applies to the concept of combination as a type of chemical bond formation, particularly in the context of valence and molecular structure.

Business and Law

In business and law, a combination refers to the joining of two or more separate companies under common ownership or control, such as a merger or acquisition. The term is used in antitrust law to describe practices that may reduce competition. A “combination in restraint of trade” is a historical legal concept derived from English common law, later codified in statutes like the Sherman Antitrust Act. Combinations can take the form of trusts, holding companies, or contractual agreements.

Other Uses

• Linguistics: The process of combining morphemes or words to form larger units, such as compounds.
• Music: A combination of notes or chords played simultaneously, often referring to harmonic progressions.
• Computing: The act of combining multiple data sets or operations, such as in combinatorial logic circuits or combination locks.

See Also

• Permutation
• Binomial coefficient
• Combination reaction
• Merger (business)
• Combinatorics