Pascal's Triangle and Binomial ExpansionsDate: 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/ |
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