Pascal’s triangle

Pascal’s triangle

Pascal’s

triangle

, also called

Tartaglia’s triangle

, is the following

configuration

of numbers:

1

1

1

1

2

1

1

3

3

1

1

4

6

4

1

1

5

10

10

5

1

1

6

15

20

15

6

1

1

7

21

35

35

21

7

1

⋮

⋮

⋮

In general, this triangle is constructed such that entries
on the left side and right side
are

1

, and every entry inside the triangle is obtained by
adding the two entries immediately above it. For

instance

,
on the fourth row

4

=

1

+

3

.

Historically, the application of this triangle has been to give
the

coefficients

when expanding

binomial

expressions

. For instance,
to expand

(

a

+

b

)

4

, one simply look up the
coefficients on the fourth row, and write

(

a

+

b

)

4

=

a

4

+

4

⁢

a

3

⁢

b

+

6

⁢

a

2

⁢

b

2

+

4

⁢

a

⁢

b

3

+

b

4

.

Pascal’s triangle is named after the French mathematician
Blaise Pascal (1623-1662)

[

3

]

.
However, this triangle was known at least
around 1100 AD in China; five centuries before Pascal

[

1

]

. In modern

language

, the expansion
of the binomial is given by
the

binomial theorem

discovered by Isaac Newton
in 1665

[

2

]

: For any

n

=

1

,

2

,

…

and

real numbers

a

,

b

, we
have

(

a

+

b

)

n

=

∑

k

=

0

n

(

n

k

)

⁢

a

n

-

k

⁢

b

k

=

a

n

+

(

n

1

)

⁢

a

n

-

1

⁢

b

+

(

n

2

)

⁢

a

n

-

2

⁢

b

2

+

⋯

+

b

n

.

Thus, in Pascal’s triangle, the entries on the

n

th row are given by
the

binomial coefficients

(

n

k

)

=

n

!

(

n

-

k

)

!

⁢

k

!

.

for

k

=

1

,

…

,

n

.

Pascal’s triangle has many interesting numerical

properties

. For example, it is easy to see that the sum of the entries in the

n

th

row is

2

n

. This can be easily proved by

induction

, but a more elegant proof goes as follows:

2

n

=

(

1

+

1

)

n

=

∑

k

=

0

n

(

n

k

)

⁢

1

n

-

k

⁢

1

k

=

∑

k

=

0

n

(

n

k

)

If you look at the long diagonals parallel to the diagonal sides of the triangle, you see in the second diagonal the

integers

1

,

2

,

3

,

4

,

…

. The next diagonal down contains the

triangular numbers

1

,

3

,

6

,

10

,

15

,

…

, and the row below that the

tetrahedral number

1

,

4

,

10

,

20

,

35

,

…

. It is easy to see why this is: for example, each triangular number is the sum of the previous triangular number and the next integer, which precisely

reflects

the

arrangement

of the triangle. Each tetrahedral number is the sum of the previous tetrahedral number and the size of the next “layer” of the tetrahedron, which is just the next triangular number. Similarly, succeeding diagonals give “triangular” number in higher

dimensions

.

References

1

Wikipedia’s

http://www.wikipedia.org/wiki/Binomial_coefficient

entry on the binomial coefficients

2

Wikipedia’s

http://www.wikipedia.org/wiki/Isaac_Newton

entry on Isaac Newton

3

Wikipedia’s

http://www.wikipedia.org/wiki/Blaise_Pascal

entry on Blaise Pascal

Title

Pascal’s triangle

Canonical

name

PascalsTriangle

Date of creation

2013-03-22 13:36:56

Last modified on

2013-03-22 13:36:56

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Topic

Classification

msc 05A10

Synonym

Tartaglia’s triangle

Related topic

BinomialCoefficient

Related topic

PascalsRule

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