# Pascal Triangle

(Redirected from Pascal's triangle)

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**See:** Triangular Structure, Pascal Triangle, Binomial Expansion, Pascal Matrix.

## References

### 2011

- (Wikipedia, 2011) ⇒ http://en.wikipedia.org/w/index.php?title=Pascal%27s_triangle&oldid=440402317
- In mathematics, '
*Pascal's triangle is a triangular array of the*n and any integer**binomial coefficients**in a triangle. … The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top. The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. This construction is related to the binomial coefficients by Pascal's rule, which states that if [math]\displaystyle{ (x+y)^n=\sum_{k=0}^n{n \choose k}x^{n-k}y^{k} }[/math] then [math]\displaystyle{ {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k} }[/math] for any nonnegative integer*k*between 0 and*n*.^{[1]}Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called*Pascal's pyramid*or*Pascal's tetrahedron*, while the general versions are called*Pascal's simplices*.

- In mathematics, '

- ↑ The binomial coefficient [math]\displaystyle{ \scriptstyle {n \choose k} }[/math] is conventionally set to zero if
*k*is either less than zero or greater than*n*.

### 2008

- (Mlodinow, 2008) ⇒ Leonard Mlodinow. (2008). “The Drunkard's Walk: How Randomness Rules Our Lives." Pantheon. ISBN: 0375424040
- QUOTE: … That was Pascal's real accomplishment: a generally applicable and systematic approach to counting that allows you to calculate the answer from a formula or read it off a chart. It is based on a curious arrangement of numbers in the shape of a triangle. The computational method at the heart of Pascal's work was actually discovered … The graphic invention employed by Pascal, given below, is thus called Pascal's triangle. In the figure, I have truncated Pascal's triangle at the tenth row, but it can be continued downward indefinitely. In fact, it is easy to continue the … Pascal's triangle is useful any time you need to know the number of ways in which you can choose some number of objects from a collection that has an equal or greater number. ...