2016 AMC 10A Problem 12

The product is odd exactly when all three selected integers are odd. There are $1008$ odd and $1008$ even integers from $1$ to $2016$. Because the integers are selected without replacement, $$p=\frac{1008}{2016}\cdot\frac{1007}{2015}\cdot\frac{1006}{2014}.$$ The first factor is $\frac12$, and each of the next two factors is slightly less than $\frac12$. Therefore $p<\frac18$. Thus, the correct answer is A.