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Binomial Expansion
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Binomial coefficient for n choose r:
(
n
r
)
=
\binom{n}{r}=
(
r
n
)
=
n
!
r
!
(
n
−
r
)
!
\frac{n!}{r!\left(n-r\right)!}
r
!
(
n
−
r
)
!
n
!
Binomial expansion formula
(
1
+
x
)
n
=
\left(1+x\right)^n=
(
1
+
x
)
n
=
1
+
1+
1
+
n
x
+
nx+
n
x
+
n
(
n
−
1
)
2
!
x
2
+
\frac{n\left(n-1\right)}{2!}x^2+
2
!
n
(
n
−
1
)
x
2
+
n
(
n
−
1
)
(
n
−
2
)
3
!
x
3
+
\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^3+
3
!
n
(
n
−
1
)
(
n
−
2
)
x
3
+
.
.
.
...
...
for
∣
x
∣
<
1
\left|x\right|<1
∣
x
∣
<
1
