In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''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, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
which can be written using
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
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
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s 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, 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 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
![Combinations without repetition; 5 choose 3](https://upload.wikimedia.org/wikipedia/commons/6/65/Combinations_without_repetition%3B_5_choose_3.svg)
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 last 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, 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
product 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.
For determining an individual binomial coefficient, it is more practical to use the formula
The
numerator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
gives the number of
''k''-permutations of ''n'', i.e., of sequences of ''k'' distinct elements of ''S'', while the
denominator 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 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, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
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 all ''k''-combinations of a given set ''S'' of ''n'' elements in some fixed order, which establishes a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 system. 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, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
:
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, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
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.
![Combinations with repetition; 5 multichoose 3](https://upload.wikimedia.org/wikipedia/commons/6/62/Combinations_with_repetition%3B_5_multichoose_3.svg)
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