COT3100BinThm01

COT3100BinThm01 - Binomial Theorem This theorem applies to...

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This theorem applies to the expansion of the expression (x+y) n . Oddly enough, this expansion is tied to the counting (in particular combinations), that we have been doing. Consider multiplying this expression out long hand: (x+y)(x+y). ....(x+y) Now, each term from this expansion is a product of terms, where each term is chosen from each binomial. One such term is x n , which is created by choosing the x in each binomial. Another term is x n-1 y. This can be created by choosing the y in the first binomial and the x in all of the following binomials. But, you could also get that same term by picking x from the first binomial, y from the second, and x from the rest. In fact, you can get that term in exactly n ways. Thus, the expansion of (x+y) n starts out as x n + nx n-1 y. It seems tedious to work this entire expansion out in this manner. We know that each term is going to be of the form x k y n-k , for integers k that range from 0 to n, inclusive. So now, the question becomes, can we get a general formula for
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COT3100BinThm01 - Binomial Theorem This theorem applies to...

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