Combination

Combination without Repetition

A combination is a way of selecting k items from a collection of n items (k ≤ n), such that (unlike permutations) the order of selection does not matter. The repetition of items is not allowed.

The number of combinations:

C_{n}^{k} = \frac{n \cdot (n - 1) \cdot (n - 2) \cdot . . . (n - k + 1)}{k!}

C_{n}^{k} = \frac{n!}{k! \cdot (n - k)!} = (\binom{n}{k})

Example:

From 5 items {a,b,c,d,e} choose 2, repetition is not allowed: $C_{5}^{2} = \frac{5!}{2! \cdot (3)!} = (\binom{5}{3}) = 10$

(a,b), (a,c), (a,d), (a,e), (b,c), (b,d), (b,e), (c,d), (c,e), (d,e)

Combination without Repetition

A combination is a way of selecting k items from a collection of n items, such that (unlike permutations) the order of selection does not matter. The repetition of items is allowed.

The number of combinations:

{\bar{C}}_{n}^{k} = (\binom{n + k - 1}{k})

Example:

From 4 items {a,b,c,d} choose 2 items, repetition is allowed:

The number of combinations: ${\bar{C}}_{4}^{2} = (\binom{4 + 2 - 1}{2}) = (\binom{5}{2}) = 10$

(a,a), (a,b), (a,c), (a,d), (b,b), (b,c), (b,d), (c,c), (c,d), (d,d)

Keywords: combination without repetition with repetition

General Information
Symbols, and relationships in mathematics
Number theory
Number Sets
Prime Numbers - Prime Factorization
Divisibility Rules
Greatest Common Divisor (factor) - Least Common Multiple
Operations with rational numbers
Binomial theorem
Binomial theorem
Combinatorics
Permutations
Combination
Variation
Sets
Operations on Sets
Cardinality of sets
Fundamental laws of set algebra
Cartesian product
Relation
Logic
Logical Operations and Truth Tables
Properties of Logical Operators
Graph Theory
Graph Theory - definitions, relationships