Abstract: The polynomial f(x) = (1/2)x^2 - (1/2)x has the following surprising property: for each integer (whole number) n, the value f(n) is also an integer, even though the coefficients are not themselves integers. We will discuss the question of identifying all the polynomials with this property. Surprisingly (or not) the key to understanding this question lies with the binomial coefficients, considered as polynomials. We will describe this relation and, time permitting, give an indication of how this gets applied to the study of multiplicities of local rings.