Pascal’s Triangle History
Binomial Coefficients
Discrete Mathematics I — MATH/COSC 1056E
Julien Dompierre
Department of Mathematics and Computer Science
Laurentian University
Sudbury, November 19, 2008, 2008
Pascal’s Triangle History
In Europe, its first record is a publication of Petrus Apianus, Rechnung
(1527). Michael Stifel (1486–1567)
and Tartaglia (1499–1557) studied it.
In Italy, it is referred to as ”Tartaglia’s
triangle”. Finally, in 1655, Blaise
Pascal wrote a Traité du triangle
arithmétique (Treatise on Arithmetical Triangle), wherein he collected
several results then known about the
triangle, and employed them to solve
problems in probability theory.
Many of the triangle’s properties were already known but not
proved. To demonstrate them, Pascal developed in his treatise an
accomplished version of the recurrence relation reasoning. He
showed the link between the triangle and the binomial theorem.
Pascal’s triangle was already known
by Persian mathematicians, for example Al-Karaji (953–1029) or Omar
Khayyam during the XIth century.
They used it to develop (a + b)n .
The triangle is referred to as the
”Khayyam triangle” in Iran. It appears in China as early as 1261 in
a book written by Yang Hui and in
Jade Mirror of the Four Unknowns
from Zhu Shijie in 1303. Yang Hui
labels it as the diagram of an ancient method first discovered by Jia
Xian before 1050. Pascal’s triangle is
called ”Yang Hui’s triangle” in China.
Historical Note: Blaise Pascal
Born June 19, 1623 in
Clermont-Ferrand, France.
Dead August 19, 1662 in
Paris.
www-groups.dcs.st-and.ac.uk/
~history/Mathematicians/
Pascal.html
Pascal’s Calculator (1642)
Powers of (x + y )
What Montreal and computers have in common?
(x
(x
(x
(x
(x
(x
(x
+ y )0
+ y )1
+ y )2
+ y )3
+ y )4
+ y )5
+ y )6
...
=
=
=
=
=
=
=
=
1
x +y
x 2 + 2xy + y 2
x 3 + 3x 2 y + 3xy 2 + y 3
x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4
x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + y 5
x 6 + 6x 5 y + 15x 4 y 2 + 20x 3 y 3 + 15x 2 y 4 + 6xy 5 + y 6
...
www.fi.muni.cz/usr/jkucera/pv109/sl1.htm
Ans.: Both began in 1642.
Pascal’s Triangle
(x
(x
(x
(x
(x
(x
(x
(x
yk
+ y )0
+ y )1
+ y )2
+ y )3
+ y )4
+ y )5
+ y )6
+ y )7
0
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
Definition: Binomial Coefficient
2
1
3
6
10
15
21
What can you observe?
3
1
4
10
20
35
4
1
5
15
35
5
1
6
21
6
7
Another usual notation to compute the number of r -combinations
of a set with n elements is
n
n!
C (n, r ) =
=
.
r
r !(n − r )!
1
7
This number is called the binomial coefficient.
1
Pascal’s Identity
yk
(x + y )0
0
0
0
1
0
2
0
3
0
4
0
5
0
6
0
7
0
(x + y )1
(x + y )2
(x + y )3
(x + y )4
(x + y )5
(x +
y )6
(x + y )7
Pascal’s Identity
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
2
2
2
3
2
4
2
5
2
6
2
7
2
3
3
3
4
3
5
3
6
3
7
3
4
4
4
5
4
6
4
7
4
5
5
5
6
5
7
5
6
6
6
7
6
7
yk
+ y )0
+ y )1
+ y )2
+ y )3
+ y )4
+ y )5
0
1
1
1
1
1
1
1
(x + y )6
(x + y )7
1
1
6
7
(x
(x
(x
(x
(x
(x
7
7
Binomial Theorem
2
1
2
3
4
5
3
4
5
1
3
1
6
4
10 + 10
ց↓
15 20
21 35
1
5
1
15
35
6
21
6
7
1
7
1
C (n + 1, k) = C (n, k − 1) + C (n, k) for 1 ≤ k ≤ n
Corollary of the Binomial Theorem
Corollary
Let n be a non negative integer. Then
Theorem
Let x and y be two variables and let n be a non negative integer.
Then
n
(x + y )
=
n
X
n
X
C (n, k) = 2n .
k=0
C (n, k)x
n−k k
y ,
k=0
n n 0
n n−1 1
n n−2 2
=
x y +
x
y +
x
y + ···
0
1
2
n
n 0 n
+
x 1 y n−1 +
x y .
n−1
n
Proof.
In the binomial theorem
n
(x + y ) =
n
X
k=0
just set x = y = 1.
C (n, k)x n−k y k ,
Corollary of the Binomial Theorem
Corollary
Let n be a non negative integer. Then
n
X
(−1)k C (n, k) = 0.
k=0
Proof.
In the binomial theorem
n
(x + y ) =
n
X
k=0
just set x = 1 and y = −1.
C (n, k)x n−k y k ,