binomial coefficient
Noun
Meaning
binomial coefficient (plural binomial coefficients)
(combinatorics) a coefficient of any of the terms in the expansion of the binomial (x+y), defined by
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
{\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}}
, read as "n choose k"
Examples
Conversely, (4) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials.
Combinatorial interpretation The binomial coefficient can be interpreted as the number of ways to choose k elements from an n-element set.
Having the indices of both rows and columns start at zero makes it possible to state that the binomial coefficient appears in the nth row and kth column of Pascal's triangle.
The same coefficient also occurs (if k ≤ n ) in the binomial formula rep rep (valid for any elements x,y of a commutative ring ), which explains the name "binomial coefficient".
For example, : The coefficient a in the term of a x b y c is known as the binomial coefficient or (the two have the same value).
In this case, the binomial coefficient : is defined when n is a real number, instead of just a positive integer.