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Pascal's Triangle and Binomial Expansions


Date: 09/01/97 at 20:43:46
From: Meghan 
Subject: College algebra

I have a question about Pascal's triangle. I need to use it to write
out the binomial expansion of something like (X+Y)^6. My book says 
there is a pattern in Pascal's triangle that you can use to find
the formula needed to solve the problem, but I can't figure out the 
pattern. If you can help me learn how to get the formula from the 
triangle I would be in heaven. Please Help!

Here is the answer to (X+Y)^6:

X^6 + 6X^5Y + 15X^4Y^2 + 20X^3Y^3 + 15X^2Y^4 + 6XY^5 + Y^6   


Date: 09/02/97 at 15:32:36
From: Doctor Anthony
Subject: Re: College algebra

The coefficients in the expansion are the numbers C(6,0), C(6,1), 
C(6,2) etc., where C(n,r) is the number of combinations of r things 
that can be made from n things.

The calculation of these coefficients is:

       C(6,0) = 1

                 6 
       C(6,1) = --- = 6
                 1

                 6.5
       C(6,2) = -----  = 15    Here 6.5 means 6 x 5
                 1.2  

                 6.5.4
       C(6,3) = -------  = 20
                 1.2.3

                 6.5.4.3
       C(6,4) =  -------  = 15
                 1.2.3.4

                 6.5.4.3.2
       C(6,5) = -----------  = 6
                 1.2.3.4.5.

                 6.5.4.3.2.1
       C(6,6) =  -----------  = 1
                 1.2.3.4.5.6  

Note the symmetry starting from either end, so C(6,0) = C(6,6)
                                               C(6,1) = C(6,5)
                                               C(6,2) = C(6,4)

We can also calculate the binomial coefficients using factorial 
notation.

                  6!
       C(6,4) = ------ 
                 4! 2!

                        n!
In general  C(n,r) = -------
                     r!(n-r)!

These would be the coefficients in the expansion of (1 + x)^n 
and C(n,r) would be the coefficient of x^r in the expansion.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Polynomials

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