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 

1 
16 
120 
560 
1820 
4368 
8008 
11440 
12870 
11440 
8008 
4368 
1820 
560 
120 
16 
1 

1 
17 
136 
680 
2380 
6188 
12376 
19448 
24310 
24310 
19448 
12376 
6188 
2380 
680 
136 
17 
1 

1 
18 
153 
816 
3060 
8568 
18564 
31824 
43758 
48620 
43758 
31824 
18564 
8568 
3060 
816 
153 
18 
1 

1 
19 
171 
969 
3876 
11628 
27132 
50388 
75582 
92378 
92378 
75582 
50388 
27132 
11628 
3876 
969 
171 
19 
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 and 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} + 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}
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 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?"
[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 


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3 

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1 

1 
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1 
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6 
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1 

1 
7 
21 
35 

35 
21 
7 
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1 
8 
28 
56 
70 
56 
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1 
9 
36 
84 
126 
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84 
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9 
1 