In the twelfth century, both Persian and Chinese mathematicians were working on a socalled arithmetic triangle that is relatively easily constructed and that gives the coefficients of the expansion of the algebraic expression (a + b)^{n} for different integer values of n (Boyer, 1991, pp. 204 and 242). Here's how it works:

Start with a row with just one entry, a 1.

Begin and end each subsequent row with a 1.

Each row should have one more entry than the row above it. Determine each interior entry by adding the number directly above the space for the new entry to the number diagonally above and to the left, so you get
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 
If you want to expand (a + b)^{10}, for example, go to the row that begins 1, 10  it's the 11th row if you start counting at 1 or the 10th row if you start counting at 0. The terms of the expansion will all be of the form a^{p}b^{q}, where p + q = 10, and p and q are whole numbers between 0 and 10. Line the terms up, starting with a^{10}b^{0}, and decreasing the power of a and increasing the power of b. The coeficients in the row are then in the proper order. So,
(a + b)^{10} = 1a^{10}b^{0 } + 10a^{9}b^{1 } + 45a^{8}b^{2 } + 120a^{7}b^{3 } + 210a^{6}b^{4 } + 252a^{5}b^{5 }
+ 210a^{4}b^{6 } + 120a^{3}b^{7 } + 45a^{2}b^{8 } + 10 a^{1}b^{9 } + 1a^{0}b^{10}.
These numbers also give the number of different ways you can choose some from a collection of objects. If you have 11 objects and want to choose 3 of them, go to the 11th row (counting from 0) and the 3rd position in (again counting from 0), and you see that there are 165 different ways to choose 3 items from a collection of 11. This brings us to Pascal.
In the mid1600s, while Blaise Pascal was working on one of his mathematical treatises, one of his friends, the Chavalier de Mere, began asking him questions about gambling odds, such as: "In eight throws of a die, a player is to attempt to throw a one, but after three unsuccessful trials, the game is interrupted. How should he be indemnified?" (Boyer and Merzbach, 1991, p. 363). Pascal's work in this area eventually led to the modern theory of probability, which has spawned the related area of statistics. Little did Pascal know where his work would lead. Nevertheless, since at the core of investigations of chance is the need to count the number of different possibilities, Pascal made use of the arithmetic triangle in his work. Because of the attention that work received, the triangle began to be known in the west as Pascal's Triangle.
The triangle is also frequently displayed in a symmetric manner where each row is centered, as in the following figure. Many people have studied the patterns to be found in the numbers in Pascal's triangle (see, for example, Brown and Hathaway, 1997; Granville, 1992, 1997; Long, 1981; and Wolfram, 1984). We will discuss one approach to looking for patterns in generalized versions of the triangle.









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 