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Combinatorial Identities Table I Intermediate Form

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the binomial theorem gives us a way to work with combinations it's useful to keep in mind that in the binomial expansion of a plus b to the N the sum of the exponents on A and B is n and this product is multiplied by n choose I so if we can rewrite a combinatorial expression in this form we can interpret it as the power of a binomial so for example let's suppose we want to find the sum 10 choose 0 plus 10 choose 1 and so on up to 10 choose 10 so let's see if we can massage these terms so they look like the terms of the binomial expansion so the binomial expansion of a plus b to the n n choose K is going to be multiplied by a power of a time a power of B where the sum of the exponents is n so remember you can get anything you want as long as you pay for it but it's also worth keeping in mind some things are free and one of those free things is multiplication by one so if we want 10 chose Z to be a power of one factor times a power of another Factor we can

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