Binomial coefficients

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 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.

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