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.