You are here

Patterns in Pascal's Triangle - with a Twist - What is Pascal's Triangle?

Author(s): 
Kathleen M. Shannon and Michael J. Bardzell

Faculty Notes 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 apbq, 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 = 1a10b 10a9b 45a8b 120a7b 210a6b 252a5b      
+ 210a4b 120a3b 45a2b 10  a1b 1a0b10.

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

JOMA

Journal of Online Mathematics and its Applications

Dummy View - NOT TO BE DELETED