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WOLFRAM|DEMONSTRATIONS PROJECT

Pascal's Triangle and the Binomial Theorem

​
n
1
11
121
1331
14641
15101051
1615201561
172135352171
0
0
1
0
1
1
2
0
2
1
2
2
3
0
3
1
3
2
3
3
4
0
4
1
4
2
4
3
4
4
5
0
5
1
5
2
5
3
5
4
5
5
6
0
6
1
6
2
6
3
6
4
6
5
6
6
7
0
7
1
7
2
7
3
7
4
7
5
7
6
7
7
7
(a+b)
7
a
+7
6
a
b+21
5
a
2
b
+35
4
a
3
b
+35
3
a
4
b
+21
2
a
5
b
+7a
6
b
+
7
b
This Demonstration illustrates the direct relation between Pascal's triangle and the binomial theorem. This very well-known connection is pointed out by the identity
n
(a+b)
=
n
0
n
a
+
n
1
n-1
a
b+
n
2
n-2
a
2
b
+…+
n
n-1
a
n-1
b
+
n
n
n
b
,n∈
, where the binomial coefficients can be obtained by using Pascal's triangle.
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