where is a binomial coefficient. This result has a nice combinatorial proof: . Repeatedly using the distributive property, we see that for a term , we must choose of the terms to contribute an to the term, and then each of the other terms of the product must contribute a . Thus, the coefficient of is . Extending this to all possible values of from to , we see that .
Consider the function for constants . It is easy to see that . Then, we have . So, the Taylor series for centered at is
Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial , one could factor it as such: . It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients.
In addition, the expansion of a polynomial such as will have coefficients corresponding to the row of Pascal's Triangle. For example, = , and the integers , , , , , and make up the 5th row of Pascal's Triangle.