Definition:Combination

Definition

Let $S$ be a set containing $n$ elements.

An $r$-combination of $S$ is a subset of $S$ which has $r$ elements.

Examples

$2$ from $3$

There are $3$ combinations of $2$ objects taken $3$ at a time:

$a \, b \quad a \, c \quad b \, c$

Also known as

A combination of elements of a set $S$ is also known as a selection from $S$.

Also see

• Cardinality of Set of Subsets: the number of $r$-combinations of $S$ is $\dfrac {n!} {r! \paren {n - r}!} = \dbinom n r$

• Definition:Combination with Repetition

• Results about combinations can be found here.

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations
• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-1}$ Permutations and Combinations: Definition $\text {3-2}$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): unordered arrangement
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): combination
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): combination
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): selection