Blaise Pascal
1623-1662



What's all the fuss and just what is Pascal's Triangle?

Pascal's triangle is an arithmetical triangle
made up of staggered rows of numbers as shown below.

 

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

    Row 0
 

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

    Row 1
 

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

    Row 2
 

 

 

 

 

  1

 

 3

 

 3

 

 1

 

 

 

 

 

    Row 3
 

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

    Row 4
 

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

    Row 5
 

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

    Row 6
 

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

    Row 7
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

    Row 8
               

.

                   
               

.

                   
               

.

                   
So How do I make my own Pascal's Triangle?
Start with the two top rows, which are always 1 and 1 1. To find any number in the next row, add the two numbers above it, as shown in the diagram below.

At the beginning and end of each row, where there is only one number above, write a 1. You can think of this rule for placing the 1's as included in the first rule: to get the first 1 in any line, you add the number above and to the left - since there is no number there, pretend it's zero - and the number above and to the right (1), to get a sum of 1.


Pascal first published his ideas on the triangle in 1665. (Although it had been known in China some centuries before this Pascal gets the credit today.) It was constructed with each horizontal line formed from the one above it by making each number equal to the sum of the numbers above and to the left in the row above. For example, the third number in the fourth line (10) equals 1 + 3 + 6.




Pascal's Triangle to Row 19

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
1     8    28    56    70    56    28     8     1
1     9    36    84   126   126    84    36     9     1
1    10    45   120   210   252   210   120    45    10     1
1    11    55   165   330   462   462   330   165    55    11     1
1    12    66   220   495   792   924   792   495   220    66    12     1
1    13    78   286   715  1287  1716  1716  1287   715   286    78    13     1
1    14    91   364  1001  2002  3003  3432  3003  2002  1001   364    91    14     1
1    15   105   455  1365  3003  5005  6435  6435  5005  3003  1365   455   105    15     1
1    16   120   560  1820  4368  8008 11440 12870 11440  8008  4368  1820   560   120    16     1
1    17   136   680  2380  6188 12376 19448 24310 24310 19448 12376  6188  2380   680   136    17     1
1    18   153   816  3060  8568 18564 31824 43758 48620 43758 31824 18564  8568  3060   816   153    18     1
1    19   171   969  3876 11628 27132 50388 75582 92378 92378 75582 50388 27132 11628  3876   969   171    19     1



Applet provided by Jeremy Baer - [email protected]
http://www.cs.washington.edu/homes/jbaer
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The natural counting numbers are easy to spot in Pascal's Triangle.
Natural Numbers in Pascal's Triangle




Triangular Numbers in Pascal's Triangle
Triangular Numbers in Pascal's Triangle



Tetrahedral Numbers in Pascal's Triangle
Tetraheydral numbers in Pascal's Triangle




Fibonacci Numbers may be found in Pascal's Triangle
Fibonacci Numbers in Pascal's Triangle




Check out lots of patterns in Pascal's Triangle
Color remainders in Pascal's Triangle

Check out this coloring of Pascal's Triangle!
Pascal's Triangle allows you to color remainders and multiples
Check out Pascal and his life
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Lots of Info about Pascal's Triangle - With graphics





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