What is pascal triangle ??????(fast)

In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after Blaise Pascal in the English-speaking world, even though others studied it centuries before him in Persia, China, India, and Italy.[1][2]
In simple terms, Pascal's triangle can be constructed in the following manner. On the first row, 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 one to the right or left is not present, substitute a zero in its place (this corresponds to the fact that
圖片參考：http://upload.wikimedia.org/math/9/a/8/9a8ba627f35ecf77e68e37092802972f.png
does not exist if k is either less than zero or greater than n). For example, the numbers 1 and 3 in the fourth row are added to produce 4 in the fifth row. More formally, this construction uses Pascal's rule, which states that


圖片參考：http://upload.wikimedia.org/math/2/c/2/2c21499ae4eea72d1be19608805c9c14.png

for positive integers n and k where
圖片參考：http://upload.wikimedia.org/math/3/c/6/3c6d5915533ef4571813e474de415fdf.png
and with the initial condition


圖片參考：http://upload.wikimedia.org/math/c/f/5/cf5fbcae49d619147c98e2af9155fe01.png

Pascal's triangle generalizes readily into higher dimensions. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron. A higher-dimensional analogue is generically called a "Pascal's simplex". See also pyramid, tetrahedron, and simplex.
For more details, pls click
http://en.wikipedia.org/wiki/Pascal's_triangle

Pascal's Triangle was originally developed by the ancient Chinese, but Blaise Pascal was the first person to discover the importance of all of the patterns it contained. On this page, I explain how the Triangle is formed, and more importantly, many of its patterns.

How Pascal's Triangle is Constructed
At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1) contains two 1's, both formed by adding the two numbers above them to the left and the right, in this case 1 and 0 (all numbers outside the Triangle are 0's). Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. In this way, the rows of the triangle go on infinitly. A number in the triangle can also be found by nCr (n Choose r) where n is the number of the row and r is the element in that row. For example, in row 3, 1 is the zeroth element, 3 is element number 1, the next three is the 2nd element, and the last 1 is the 3rd element. The formula for nCr is:
n!
--------
r!(n-r)!

! means factorial, or the preceeding number multiplied by all the positive integers that are smaller than the number. 5! = 5 × 4 × 3 × 2 × 1 = 120.

The Sums of the Rows
The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row. For example:


20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16

Prime Numbers
If the 1st element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1's) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.

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


巴斯卡是十七世紀的一位法國數學家，也是歷史上第一位發明了加法計算機的人！他造出「巴斯卡三角形」的方法是這樣的：先在紙上寫出一行和一列的「１」，然後在各個位置中填入數字，每一個位置上的數字都是它上面一個數和左邊一個數的和。接下來，把這個表右轉 45°，放正了，就得到上面的數字三角形了！

其實，這個三角形的每一列數字，剛好就是中學會學到的 (a+b)n的展開式的係數表：

(a+b)0 = 1
1

(a+b)1 = a+b
1 1

(a+b)2 = a2+2ab+b2
1 2 1

(a+b)3 = a3+3a2b+3ab2+b3
1 3 3 1

(a+b)4 = a4+4a3b+6a2b2+4ab3+b4
1 4 6 4 1

(a+b)5 = a5+5a4b+10a3b2+10a2b3+5ab4+b5
1 5 10 10 5 1

(a+b)6 = a6+6a5b+15a4b2+20a3b3+15a2b4+6ab5+b6
1 6 15 20 15 6 1



http://mathforum.org/dr.math/faq/faq.pascal.triangle.html <-英文解釋