Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example (for n = 4),




(
x
+
y

)

4


=

x

4


+
4

x

3


y
+
6

x

2



y

2


+
4
x

y

3


+

y

4


.


{\displaystyle (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 binomial coefficient







(


n
b


)






{\displaystyle {\tbinom {n}{b}}}
or







(


n
c


)






{\displaystyle {\tbinom {n}{c}}}
(the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where







(


n
b


)






{\displaystyle {\tbinom {n}{b}}}
gives the number of different combinations of b elements that can be chosen from an n-element set. Therefore







(


n
b


)






{\displaystyle {\tbinom {n}{b}}}
is often pronounced as "n choose b".

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