Combinatorics is the study of the compilation of countably many objects. This is generally the number of possibilities for a certain composition in the foreground, as it can be derived a statement about the probability of a particular compilation. One differentiates by the compilations between permutations, variations and combinations of objects.
List of the basic functions for combinatorics.
The function f(n) = n! with f(0) = 1 and f(n) = n ⋅ f(n-1) is called n-Faculty.
Example: 5! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 = 120
The defined for all natural numbers function
is called a binomial coefficient.
The binomial coefficient is defined as follows:
Properties of the binomial coefficients
For all real numbers a and b and all natural numbers n holds the binomial theorem:
Calculator: binomial coefficient
Calculator: Binomial theorem
The binomial coefficients are a special case of the more general polynomial coefficients. The polynomial coefficients are defined for all natural n and all r-tuples of natural ki for which the sum is equal to n.
The polynomial coefficients are defined as follows:
For all r-tuples of real numbers a 1 , a 2 , ..., a r and all natural numbers n we have the polynomial theorem. Here the sum of all the combinations of r-tuples is to be formed for which the sum is equal to n.
As a permutation P k is called the sequence of k elements considering the position of the elements in the series. A question of combinatorics is how many orders A of k elements are possible.
As a variation V k r without repetition is called the ordered selection of r elements from a set of k elements. The difference to combinations is the considering of the order of the elements. For example, if participate in a tournament k teams is the number of possible first three places a variation without repetitions as each team can only occupy a place and position in the top three is relevant.
It applies to variations without repetitions:
It applies to variations with repetitions:
As a combination of C k r is defined as the selection of r elements from a set of k elements. In contrast to the variations, the order does not matter. For example, the lottery numbers are a combination without repetition, because the drawn numbers will not back down.
It applies to combinations without repetitions:
It applies to combinations with repetitions: