Proposition: For any real numbers
a
,
b
{\displaystyle a,b}
n
≥
0
,
{\displaystyle n\geq 0,}
(
a
+
b
)
n
=
∑
k
=
0
n
(
n
k
)
a
n
−
k
b
k
{\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k}}
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}
Proof (Mathematical Induction) :
For
n
=
0
,
(
a
+
b
)
n
=
(
a
+
b
)
0
=
1
{\displaystyle n=0,(a+b)^{n}=(a+b)^{0}=1}
For
n
=
1
,
(
a
+
b
)
n
=
(
a
+
b
)
1
=
(
a
+
b
)
{\displaystyle n=1,(a+b)^{n}=(a+b)^{1}=(a+b)}
n
=
2
,
(
a
+
b
)
n
=
(
a
+
b
)
2
{\displaystyle n=2,(a+b)^{n}=(a+b)^{2}}
=
a
2
+
2
a
b
+
b
2
{\displaystyle =a^{2}+2ab+b^{2}}
=
(
2
0
)
a
2
b
0
+
(
2
1
)
a
1
b
1
+
(
2
2
)
a
0
b
2
{\displaystyle ={\binom {2}{0}}a^{2}b^{0}+{\binom {2}{1}}a^{1}b^{1}+{\binom {2}{2}}a^{0}b^{2}}
=
∑
k
=
0
2
(
2
k
)
a
2
−
k
b
k
{\displaystyle =\sum _{k=0}^{2}{\binom {2}{k}}a^{2-k}b^{k}}
n
=
3
,
(
a
+
b
)
n
=
(
a
+
b
)
3
{\displaystyle n=3,(a+b)^{n}=(a+b)^{3}}
=
a
3
+
3
a
2
b
+
3
a
b
2
+
b
3
{\displaystyle =a^{3}+3a^{2}b+3ab^{2}+b^{3}}
=
(
3
0
)
a
3
b
0
+
(
3
1
)
a
2
b
1
+
(
3
2
)
a
1
b
2
+
(
3
3
)
a
0
b
3
{\displaystyle ={\binom {3}{0}}a^{3}b^{0}+{\binom {3}{1}}a^{2}b^{1}+{\binom {3}{2}}a^{1}b^{2}+{\binom {3}{3}}a^{0}b^{3}}
=
∑
k
=
0
3
(
3
k
)
a
3
−
k
b
k
{\displaystyle =\sum _{k=0}^{3}{\binom {3}{k}}a^{3-k}b^{k}}
(
a
+
b
)
n
=
∑
k
=
0
n
(
n
k
)
a
n
−
k
b
k
{\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k}}
a
≥
0.
{\displaystyle a\geq 0.}
n
+
1.
{\displaystyle n+1.}
(
a
+
b
)
n
=
∑
k
=
0
n
(
n
k
)
a
n
−
k
b
k
{\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k}}
Or,
(
a
+
b
)
n
(
a
+
b
)
=
(
∑
k
=
0
n
(
n
k
)
a
n
−
k
b
k
)
(
a
+
b
)
{\displaystyle (a+b)^{n}(a+b)=(\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k})(a+b)}
=
(
(
n
0
)
a
n
b
0
+
(
n
1
)
a
n
−
1
b
1
+
(
n
2
)
a
n
−
2
b
2
+
.
.
.
+
(
n
n
)
a
0
b
n
)
(
a
+
b
)
{\displaystyle =({\binom {n}{0}}a^{n}b^{0}+{\binom {n}{1}}a^{n-1}b^{1}+{\binom {n}{2}}a^{n-2}b^{2}+...+{\binom {n}{n}}a^{0}b^{n})(a+b)}
=
a
(
(
n
0
)
a
n
b
0
+
(
n
1
)
a
n
−
1
b
1
+
(
n
2
)
a
n
−
2
b
2
+
.
.
.
+
(
n
n
)
a
0
b
n
)
+
b
(
(
n
0
)
a
n
b
0
+
(
n
1
)
a
n
−
1
b
1
+
(
n
2
)
a
n
−
2
b
2
+
.
.
.
+
(
n
n
)
a
0
b
n
)
{\displaystyle =a({\binom {n}{0}}a^{n}b^{0}+{\binom {n}{1}}a^{n-1}b^{1}+{\binom {n}{2}}a^{n-2}b^{2}+...+{\binom {n}{n}}a^{0}b^{n})+b({\binom {n}{0}}a^{n}b^{0}+{\binom {n}{1}}a^{n-1}b^{1}+{\binom {n}{2}}a^{n-2}b^{2}+...+{\binom {n}{n}}a^{0}b^{n})}
=
(
(
n
0
)
a
n
+
1
b
0
+
(
n
1
)
a
n
b
1
+
(
n
2
)
a
n
−
1
b
2
+
.
.
.
+
(
n
n
)
a
b
n
)
+
(
(
n
0
)
a
n
b
1
+
(
n
1
)
a
n
−
1
b
2
+
(
n
2
)
a
n
−
2
b
3
+
.
.
.
+
(
n
n
)
a
0
b
n
+
1
)
{\displaystyle =({\binom {n}{0}}a^{n+1}b^{0}+{\binom {n}{1}}a^{n}b^{1}+{\binom {n}{2}}a^{n-1}b^{2}+...+{\binom {n}{n}}ab^{n})+({\binom {n}{0}}a^{n}b^{1}+{\binom {n}{1}}a^{n-1}b^{2}+{\binom {n}{2}}a^{n-2}b^{3}+...+{\binom {n}{n}}a^{0}b^{n+1})}
=
(
n
0
)
a
n
+
1
b
0
+
a
n
b
1
(
(
n
1
)
+
(
n
0
)
)
+
a
n
−
1
b
2
(
(
n
2
)
+
(
n
1
)
)
+
.
.
.
+
a
b
n
(
(
n
n
)
+
(
n
n
−
1
)
)
+
(
n
n
)
a
0
b
n
+
1
)
{\displaystyle ={\binom {n}{0}}a^{n+1}b^{0}+a^{n}b^{1}({\binom {n}{1}}+{\binom {n}{0}})+a^{n-1}b^{2}({\binom {n}{2}}+{\binom {n}{1}})+...+ab^{n}({\binom {n}{n}}+{\binom {n}{n-1}})+{\binom {n}{n}}a^{0}b^{n+1})}
=
(
n
+
1
0
)
a
n
+
1
b
0
+
(
n
+
1
1
)
a
n
b
1
+
(
n
+
1
2
)
a
n
−
1
b
2
+
.
.
.
+
(
n
+
1
n
)
a
b
n
+
(
n
+
1
n
+
1
)
a
0
b
n
+
1
)
{\displaystyle ={\binom {n+1}{0}}a^{n+1}b^{0}+{\binom {n+1}{1}}a^{n}b^{1}+{\binom {n+1}{2}}a^{n-1}b^{2}+...+{\binom {n+1}{n}}ab^{n}+{\binom {n+1}{n+1}}a^{0}b^{n+1})}
∴
(
a
+
b
)
n
+
1
=
∑
k
=
0
n
+
1
(
n
+
1
k
)
a
n
+
1
−
k
b
k
{\displaystyle (a+b)^{n+1}=\sum _{k=0}^{n+1}{\binom {n+1}{k}}a^{n+1-k}b^{k}}
