In
mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike
permutation
In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

s). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a ''k''-combination of a
set ''S'' is a subset of ''k'' distinct elements of ''S''. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has ''n'' elements, the number of ''k''-combinations, denoted as
, is equal to the
binomial coefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
:
which can be written using
factorial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s as
whenever
, and which is zero when
. This formula can be derived from the fact that each ''k''-combination of a set ''S'' of ''n'' members has
permutations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

so
or
. The set of all ''k''-combinations of a set ''S'' is often denoted by
.
A combination is a combination of ''n'' things taken ''k'' at a time ''without repetition''. To refer to combinations in which repetition is allowed, the terms ''k''-selection, ''k''-
multiset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, or ''k''-combination with repetition are often used. If, in the above example, it were possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears.
Although the set of three fruits was small enough to write a complete list of combinations, this becomes impractical as the size of the set increases. For example, a
poker hand
In poker
Poker is a family of card games in which Card player, players betting (poker), wager over which poker hand, hand is best according to that specific game's rules in ways similar List of poker hands, to these rankings. While the earli ...
can be described as a 5-combination (''k'' = 5) of cards from a 52 card deck (''n'' = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.
Number of ''k''-combinations

The
number of ''k''-combinations from a given set ''S'' of ''n'' elements is often denoted in elementary combinatorics texts by
, or by a variation such as
,
,
,
or even
(the latter form is standard in French, Romanian, Russian, Chinese and Polish texts). The same number however occurs in many other mathematical contexts, where it is denoted by
(often read as "''n'' choose ''k''"); notably it occurs as a coefficient in the
binomial formula
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of exponentiation, powers of a binomial (polynomial), binomial. According to the theorem, it is possible to expand the polynomial into a summati ...
, hence its name binomial coefficient. One can define
for all natural numbers ''k'' at once by the relation
:
from which it is clear that
:
and further,
:
for ''k'' > ''n''.
To see that these coefficients count ''k''-combinations from ''S'', one can first consider a collection of ''n'' distinct variables ''X''
''s'' labeled by the elements ''s'' of ''S'', and expand the
over all elements of ''S'':
:
it has 2
''n'' distinct terms corresponding to all the subsets of ''S'', each subset giving the product of the corresponding variables ''X''
''s''. Now setting all of the ''X''
''s'' equal to the unlabeled variable ''X'', so that the product becomes , the term for each ''k''-combination from ''S'' becomes ''X''
''k'', so that the coefficient of that power in the result equals the number of such ''k''-combinations.
Binomial coefficients can be computed explicitly in various ways. To get all of them for the expansions up to , one can use (in addition to the basic cases already given) the recursion relation
:
for 0 < ''k'' < ''n'', which follows from =; this leads to the construction of
Pascal's triangle
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
.
For determining an individual binomial coefficient, it is more practical to use the formula
:
.
The
numerator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
gives the number of
''k''-permutations of ''n'', i.e., of sequences of ''k'' distinct elements of ''S'', while the
denominator
A fraction (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
gives the number of such ''k''-permutations that give the same ''k''-combination when the order is ignored.
When ''k'' exceeds ''n''/2, the above formula contains factors common to the numerator and the denominator, and canceling them out gives the relation
:
for 0 ≤ ''k'' ≤ ''n''. This expresses a symmetry that is evident from the binomial formula, and can also be understood in terms of ''k''-combinations by taking the
complement
A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be:
* Complement (linguistics), a word or phrase having a particular syntactic role
** Subject complement, a word or phrase addi ...
of such a combination, which is an -combination.
Finally there is a formula which exhibits this symmetry directly, and has the merit of being easy to remember:
:
where ''n''
! denotes the
factorial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
of ''n''. It is obtained from the previous formula by multiplying denominator and numerator by !, so it is certainly computationally less efficient than that formula.
The last formula can be understood directly, by considering the ''n''
! permutations of all the elements of ''S''. Each such permutation gives a ''k''-combination by selecting its first ''k'' elements. There are many duplicate selections: any combined permutation of the first ''k'' elements among each other, and of the final (''n'' − ''k'') elements among each other produces the same combination; this explains the division in the formula.
From the above formulas follow relations between adjacent numbers in Pascal's triangle in all three directions:
:
.
Together with the basic cases
, these allow successive computation of respectively all numbers of combinations from the same set (a row in Pascal's triangle), of ''k''-combinations of sets of growing sizes, and of combinations with a complement of fixed size .
Example of counting combinations
As a specific example, one can compute the number of five-card hands possible from a standard fifty-two card deck as:
:
Alternatively one may use the formula in terms of factorials and cancel the factors in the numerator against parts of the factors in the denominator, after which only multiplication of the remaining factors is required:
:
Another alternative computation, equivalent to the first, is based on writing
:
which gives
:
.
When evaluated in the following order, , this can be computed using only integer arithmetic. The reason is that when each division occurs, the intermediate result that is produced is itself a binomial coefficient, so no remainders ever occur.
Using the symmetric formula in terms of factorials without performing simplifications gives a rather extensive calculation:
:
Enumerating ''k''-combinations
One can
enumerate
Enumerate could refer to:
* Enumeration, a mathematical, theoretical concept of an exhaustive listing of compatible items
* Enumerate (project), a collaborative research project about digitization of cultural heritage
* Enumerated type in computer p ...
all ''k''-combinations of a given set ''S'' of ''n'' elements in some fixed order, which establishes a
bijection
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from an interval of
integers with the set of those ''k''-combinations. Assuming ''S'' is itself ordered, for instance ''S'' = , there are two natural possibilities for ordering its ''k''-combinations: by comparing their smallest elements first (as in the illustrations above) or by comparing their largest elements first. The latter option has the advantage that adding a new largest element to ''S'' will not change the initial part of the enumeration, but just add the new ''k''-combinations of the larger set after the previous ones. Repeating this process, the enumeration can be extended indefinitely with ''k''-combinations of ever larger sets. If moreover the intervals of the integers are taken to start at 0, then the ''k''-combination at a given place ''i'' in the enumeration can be computed easily from ''i'', and the bijection so obtained is known as the
combinatorial number systemIn mathematics, and in particular in combinatorics, the combinatorial number system of degree ''k'' (for some positive integer ''k''), also referred to as combinadics, is a correspondence between natural numbers (taken to include 0) ''N'' and '' ...
. It is also known as "rank"/"ranking" and "unranking" in computational mathematics.
There are many ways to enumerate ''k'' combinations. One way is to visit all the binary numbers less than 2
''n''. Choose those numbers having ''k'' nonzero bits, although this is very inefficient even for small ''n'' (e.g. ''n'' = 20 would require visiting about one million numbers while the maximum number of allowed ''k'' combinations is about 186 thousand for ''k'' = 10). The positions of these 1 bits in such a number is a specific ''k''-combination of the set . Another simple, faster way is to track ''k'' index numbers of the elements selected, starting with (zero-based) or (one-based) as the first allowed ''k''-combination and then repeatedly moving to the next allowed ''k''-combination by incrementing the last index number if it is lower than ''n''-1 (zero-based) or ''n'' (one-based) or the last index number ''x'' that is less than the index number following it minus one if such an index exists and resetting the index numbers after ''x'' to .
Number of combinations with repetition
A ''k''-combination with repetitions, or ''k''-multicombination, or
multisubset of size ''k'' from a set ''S'' of size ''n'' is given by a set of ''k'' not necessarily distinct elements of ''S'', where order is not taken into account: two sequences define the same multiset if one can be obtained from the other by permuting the terms. In other words, it is a sample of ''k'' elements from a set of ''n'' elements allowing for duplicates (i.e., with replacement) but disregarding different orderings (e.g. = ). Associate an index to each element of ''S'' and think of the elements of ''S'' as ''types'' of objects, then we can let
denote the number of elements of type ''i'' in a multisubset. The number of multisubsets of size ''k'' is then the number of nonnegative integer (so allowing zero) solutions of the
Diophantine equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
:
:
If ''S'' has ''n'' elements, the number of such ''k''-multisubsets is denoted by,
:
a notation that is analogous to the
binomial coefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
which counts ''k''-subsets. This expression, ''n'' multichoose ''k'', can also be given in terms of binomial coefficients:
:
This relationship can be easily proved using a representation known as
stars and bars.
A solution of the above Diophantine equation can be represented by
''stars'', a separator (a ''bar''), then
more stars, another separator, and so on. The total number of stars in this representation is ''k'' and the number of bars is ''n'' - 1 (since a separation into n parts needs n-1 separators). Thus, a string of ''k'' + ''n'' - 1 (or ''n'' + ''k'' - 1) symbols (stars and bars) corresponds to a solution if there are ''k'' stars in the string. Any solution can be represented by choosing ''k'' out of positions to place stars and filling the remaining positions with bars. For example, the solution
of the equation
(''n'' = 4 and ''k'' = 10) can be represented by
The number of such strings is the number of ways to place 10 stars in 13 positions,
which is the number of 10-multisubsets of a set with 4 elements.

As with binomial coefficients, there are several relationships between these multichoose expressions. For example, for
,
:
This identity follows from interchanging the stars and bars in the above representation.
Example of counting multisubsets
For example, if you have four types of donuts (''n'' = 4) on a menu to choose from and you want three donuts (''k'' = 3), the number of ways to choose the donuts with repetition can be calculated as
:
This result can be verified by listing all the 3-multisubsets of the set ''S'' = . This is displayed in the following table. The second column lists the donuts you actually chose, the third column shows the nonnegative integer solutions