Edit made on January 03, 2015 by ColinWright at 18:52:45
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HEADERS_END
[[[> |>>
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 _
/etc/
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Pascal's Triangle
<<| ]]]
Each number in Pascal's Triangle is the sum of the number two numbers directly above it and the number to the immediate left of that number.
above. The start of Pascal's Triangle is shown below, here, where the top row,
containing just one 1, is row 0.
| 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 |
Pascal's Triangle is very important for binomial expansions. expansions, the contents
being the binomial coefficients.
The notations used for a number in Pascal's Triangle are:
* EQN:\left(n\\r\right) $n\choose~r$
* EQN:_^nC_r ${}^nC_r$
where EQN:n $n$ is the row number and EQN:r $r$ is the EQN:r^{th} $r^{th}$ term in the EQN:n^{th} $n^{th}$ row
e.g. EQN:\left(4\\3\right)=6 ${4\choose~2}~=~6$
The formula for finding any number in Pascal's Triangle is:
* EQN:\left(n\\r\right)=\frac{n!}{r!(n-r)!} ${n\choose~r}~=~\frac{n!}{r!(n-r)!}$
where EQN:n! $n!$ is n $n$ factorial
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See http://mathworld.wolfram.com/PascalsTriangle.html for more information.