In the twelfth century, both Persian and Chinese mathematicians were working on a so-called *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} = **1***a*^{10}*b*^{0 } *+ * **10***a*^{9}*b*^{1 } *+ * **45***a*^{8}*b*^{2 } *+ * **120***a*^{7}*b*^{3 } *+ * **210***a*^{6}*b*^{4 } *+ * **252***a*^{5}*b*^{5 }

*+* **210***a*^{4}*b*^{6 } *+ * **120***a*^{3}*b*^{7 } *+ * **45***a*^{2}*b*^{8 } *+ * **10 ** *a*^{1}*b*^{9 } *+ * **1***a*^{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 mid-1600s, 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 |

Kathleen M. Shannon and Michael J. Bardzell, "Patterns in Pascal's Triangle - with a Twist - What is Pascal's Triangle?," *Loci* (December 2004)

## JOMA

Journal of Online Mathematics and its Applications