"nCk" redirects here. For other uses, see NCK (disambiguation).
The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above.Visualisation of binomial expansion up to the 4th power
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 n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula
which using factorial notation can be compactly expressed as
For example, the fourth power of 1 + x is
and the binomial coefficient is the coefficient of the x2 term.
Arranging the numbers in successive rows for n = 0, 1, 2, ... gives a triangular array called Pascal's triangle, satisfying the recurrence relation
The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol is usually read as "n choose k" because there are ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are ways to choose 2 elements from {1, 2, 3, 4}, namely {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4} and {3, 4}.
The binomial coefficients can be generalized to for any complex number z and integer k ≥ 0, and many of their properties continue to hold in this more general form.
and 21 Related for: Binomial coefficient information
mathematics, the binomialcoefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomialcoefficient is indexed...
In mathematics the nth central binomialcoefficient is the particular binomialcoefficient ( 2 n n ) = ( 2 n ) ! ( n ! ) 2 = ∏ k = 1 n n + k k for all ...
Gaussian binomialcoefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomialcoefficients) are q-analogs of the binomial coefficients...
(x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.} The coefficient a in the term of axbyc is known as the binomialcoefficient ( n b ) {\displaystyle {\tbinom {n}{b}}}...
! {\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}} is the binomialcoefficient, hence the name of the distribution. The formula can be understood...
right-hand side of (1) is expressed in terms of the (generalized) binomialcoefficients ( α k ) := α ( α − 1 ) ( α − 2 ) ⋯ ( α − k + 1 ) k ! . {\displaystyle...
positive covariance term. The term "negative binomial" is likely due to the fact that a certain binomialcoefficient that appears in the formula for the probability...
Look up binomial in Wiktionary, the free dictionary. Binomial may refer to: Binomial (polynomial), a polynomial with two terms Binomialcoefficient, numbers...
Like the binomial distribution that involves binomialcoefficients, there is a negative binomial distribution in which the multiset coefficients occur....
{\displaystyle C(n,k)} or C k n {\displaystyle C_{k}^{n}} , is equal to the binomialcoefficient ( n k ) = n ( n − 1 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ⋯ 1 , {\displaystyle...
v=x_{1}e_{1}+x_{2}e_{2}+\dotsb +x_{n}e_{n}.} Correlation coefficient Degree of a polynomial Monic polynomial Binomialcoefficient "ISO 80000-1:2009". International Organization...
partition yields a partition of n − M into at most M parts. The Gaussian binomialcoefficient is defined as: ( k + ℓ ℓ ) q = ( k + ℓ k ) q = ∏ j = 1 k + ℓ ( 1...
Bernoulli number, and ( p k ) {\displaystyle {\binom {p}{k}}} is a binomialcoefficient. In the following summations, a is assumed to be different from 1...
_{s}(z)} is a polylogarithm. ( n k ) {\displaystyle n \choose k} is binomialcoefficient exp ( x ) {\displaystyle \exp(x)} denotes exponential of x {\displaystyle...
function for binomialcoefficients for a fixed n, one may ask for a bivariate generating function that generates the binomialcoefficients (n k) for all...
powerful insights into each or both of the sets. The symmetry of the binomialcoefficients states that ( n k ) = ( n n − k ) . {\displaystyle {n \choose k}={n...
n − 1 ) 2 {\displaystyle {n \choose 2}={n(n-1) \over 2}} is the binomialcoefficient for the number of ways to choose two items from n items. The number...
theorem are the multinomial coefficients. They can be expressed in numerous ways, including as a product of binomialcoefficients or of factorials: ( n k...
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