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General Information, Facts & Tables
[Pascal's Triangle]

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Overview

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Blaise Pascal (1623 - 1662), a French mathematician, physicist, and philosopher discovered that when a specific sequence of numbers are arranged in a triangular format they relate to the certain probability ratios. The triangle is created by starting with a one at the top position, then each successive line is created by placing a one at either end of the row, and the value of the numbers towards the centre of the row are derived from the sum of the two numbers directly above it.
 

Pascal's Triangle (First 7 Rows)

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(a + b)0

 

 

 

 

 

 

1

 

 

 

 

 

 

(a + b)1

 

 

 

 

 

1

 

1

 

 

 

 

 

(a + b)2

 

 

 

 

1

 

2

 

1

 

 

 

 

(a + b)3

 

 

 

1

 

3

 

3

 

1

 

 

 

(a + b)4

 

 

1

 

4

 

6

 

4

 

1

 

 

(a + b)5

 

1

 

5

 

10

 

10

 

5

 

1

 

(a + b)6

1

 

6

 

15

 

20

 

15

 

6

 

1

 

Rules

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  • Each line begins with one and ends with one.
  • When the digits of each row are placed beside each other, the numeric value is a power of 11. eg 11 = 111, 121 = 112, 1331= 113.
  • The sum of all of the numbers in a row is a power of 2. eg 1+1=2=21, 1+2+1 = 4 = 22, 1+3+3+1 = 8 = 23.
  • Each row shows the coefficients for (a + b)n. eg (a + b)5 = "1, 5, 10, 10, 5, 1" = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5.