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
(
n
k
)
.
{\displaystyle {\tbinom {n}{k}}.}
It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
.
{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}
For example, the fourth power of 1 + x is
(
1
+
x
)
4
=
(
4
0
)
x
0
+
(
4
1
)
x
1
+
(
4
2
)
x
2
+
(
4
3
)
x
3
+
(
4
4
)
x
4
=
1
+
4
x
+
6
x
2
+
4
x
3
+
x
4
,
{\displaystyle {\begin{aligned}(1+x)^{4}&={\tbinom {4}{0}}x^{0}+{\tbinom {4}{1}}x^{1}+{\tbinom {4}{2}}x^{2}+{\tbinom {4}{3}}x^{3}+{\tbinom {4}{4}}x^{4}\\&=1+4x+6x^{2}+4x^{3}+x^{4},\end{aligned}}}
and the binomial coefficient
(
4
2
)
=
4
!
2
!
2
!
=
6
{\displaystyle {\tbinom {4}{2}}={\tfrac {4!}{2!2!}}=6}
is the coefficient of the x2 term.
Arranging the numbers
(
n
0
)
,
(
n
1
)
,
…
,
(
n
n
)
{\displaystyle {\tbinom {n}{0}},{\tbinom {n}{1}},\ldots ,{\tbinom {n}{n}}}
in successive rows for
n
=
0
,
1
,
2
,
…
{\displaystyle n=0,1,2,\ldots }
gives a triangular array called Pascal's triangle, satisfying the recurrence relation
(
n
k
)
=
(
n
−
1
k
)
+
(
n
−
1
k
−
1
)
.
{\displaystyle {\binom {n}{k}}={\binom {n-1}{k}}+{\binom {n-1}{k-1}}.}
The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol
(
n
k
)
{\displaystyle {\tbinom {n}{k}}}
is usually read as "n choose k" because there are
(
n
k
)
{\displaystyle {\tbinom {n}{k}}}
ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are
(
4
2
)
=
6
{\displaystyle {\tbinom {4}{2}}=6}
ways to choose 2 elements from
{
1
,
2
,
3
,
4
}
,
{\displaystyle \{1,2,3,4\},}
namely
{
1
,
2
}
,
{
1
,
3
}
,
{
1
,
4
}
,
{
2
,
3
}
,
{
2
,
4
}
,
{\displaystyle \{1,2\}{\text{, }}\{1,3\}{\text{, }}\{1,4\}{\text{, }}\{2,3\}{\text{, }}\{2,4\}{\text{,}}}
and
{
3
,
4
}
.
{\displaystyle \{3,4\}.}
The binomial coefficients can be generalized to
(
z
k
)
{\displaystyle {\tbinom {z}{k}}}
for any complex number z and integer k ≥ 0, and many of their properties continue to hold in this more general form.
View More On Wikipedia.org