# binomial coefficient

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Related to binomial coefficient: binomial theorem, binomial distribution

## binomial coefficient

[bī′nō·mē·əl kō·ə′fish·ənt]
(mathematics)
A coefficient in the expansion of (x + y) n , where n is a positive integer; the (k + 1)st coefficient is equal to the number of ways of choosing k objects out of n without regard for order. Symbolized (nk); nCk ; C (n,k); Cnk.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
The binomial coefficient of (4) can be generalized to real arguments by means of the Euler's gamma function:
The computer program for Equations (12), (16), (17), and (19) containing simple finite sums of binomial coefficients was developed using Mathematica 8.0 software.
All other negative binomial coefficients presented in table 2 are of the anticipated signs and are significantly different from zero with the exception of the coefficients for the destination country population variable.
Using similar techniques, we prove that the number of 312-avoiding affine permutations in [[??].sub.d] with k cut-points is given by the binomial coefficient [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
We give the name Pascal descent polynomial to [p.sub.n] (I) since it yields a signed binomial coefficient when [absolute value of I] = 1 and it originates from the additive formula (4.3) which relates to the descent sums [D.sub.[k-1], for k [member of] [n].
The binomial coefficients in the expression (12) for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are now of the form
Remark 1: Also note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is a binomial coefficient for each n [greater than or equal to] 1.
Note that using the general binomial coefficient one can write (11) as
An important tool for counting subwords is the notion of binomial coefficient of words, a generalization of the classical binomial coefficient for integers.
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the binomial coefficient.
On the residue of a binomial coefficient with respect to a prime modulus.
However if we classify our choices by the value m + 1 of [x.sub.a + b + 1], then we see that ([.sup.m.sub.a]) can be viewed as the number of ways to pick the circled points from [m], ([??]) can be viewed as the number of ways to pick the non-circled chosen points from [m], and the binomial coefficient ([??]) can be viewed as the number of ways to pick the points [x.sub.a + b + 2] < ...

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