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Alternative notations include C(n, k), n C k, n C k, C k n, [3] C n k, and C n,k, in all of which the C stands for combinations or choices; the C notation means the number of ways to choose k out of n objects. Many calculators use variants of the C notation because they can represent it on a single-line display.
There are () paths starting at (0, 0) that end at (k, m−k), as k right moves and m−k upward moves must be made (and the path length is m). Similarly, there are ( n r − k ) {\displaystyle {\binom {n}{r-k}}} paths starting at ( k , m − k ) that end at ( r , m + n − r ), as a total of r − k right moves and ( m + n − r ) − ( m − k ...
In 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 permutations).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.
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.
To see the expression (+) directly, observe that any arrangement of stars and bars consists of a total of n + k − 1 symbols, n of which are stars and k − 1 of which are bars. Thus, we may lay out n + k − 1 slots and choose k − 1 of these to contain bars (or, equivalently, choose n of the slots to contain stars).
The binomial distribution is the PMF of k successes given n independent events each with a probability p of success. Mathematically, when α = k + 1 and β = n − k + 1, the beta distribution and the binomial distribution are related by [clarification needed] a factor of n + 1:
The Stirling numbers of the second kind, S(n,k) count the number of partitions of a set of n elements into k non-empty subsets (indistinguishable boxes). An explicit formula for them can be obtained by applying the principle of inclusion–exclusion to a very closely related problem, namely, counting the number of partitions of an n -set into k ...
In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations.