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PASCAL TRIANGLE

The Pascal Triangle is an arrangement of numbers in the form of a triangle in such a manner that each number on the triangle is equal to the sum of the two numbers above it.  This triangle is named after Blaise Pascal (1623-1662), a French mathematician who studied it extensively.  However, other mathematicians in Persia and China may have studied it centuries before him.

Below is the Pascal 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        

where the first eight rows are shown.  The triangle can be of enumerated for any size and may go into infinity.

It can be noted that the upper left and upper right sides of the triangle are composed of the numeral 1.  The other elements of the triangle are constructed by placing into the respective positions of the triangle a number equal to the sum of the two numbers above it.

One usefulness of the Pascal Triangle can be seen on the coefficients of the binomial expansion of the term

(x + y)n

For example, for n=5, the binomial expansion of  (x + y)5 is:

(1)x5 + (5)x4y1 + (10)x3y2 + (10)x2y3 + (5)x1y4 + (1)y5

where it can be noted that the coefficients of the expansion corresponds to the elements of a row in the Pascal Triangle.  In this case, this is the 6th row of the Pascal Triangle.

 

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