What is Pascal's Triangle?

• Start with a row with just one entry, a one.
• Begin and end each subsequent row with a one.
• Each row should have one more entry than the one 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:

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 and the 10th row if you start counting at 0). The terms of the expansion will all be of the form a^pb^q where p+q=10 and p and q are whole numbers between 0 and 10. Line the terms up starting with a¹⁰b⁰ 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 = 1a¹⁰b⁰ + 10 a⁹b¹ +45 a⁸b² +120 a⁷b³ + 210 a⁶b⁴ +252 a⁵b⁵ +210 a⁴b⁶ +120 a³b⁷ + 45 a²b⁸ +10 a¹b⁹ +1 a⁰b¹⁰
Remember that anything raised to the zero power is 1.
        Another application of these numbers is that they give the number of different ways you can choose some of a collection of objects. If you have 11 objects and you want to choose 3 of them, go to the 11th row (start counting at 0) and the 3rd position in, again starting to count at 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?" [3, pg 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 below. Numerous people have studied the patterns to be found in the numbers in Pascal's triangle. (See for example [4],[8],[9],[11] and [14].) In this paper 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		1		1		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