Suppose you have 500 elements and select a group of 20 out of them. What is the probability of any element being in the group?

I've come up with an answer two different ways, but both of them disagree with the book's answer. The book says that it's 1/500.

However, I would think that I'd need to calculate $\displaystyle \frac{\binom{499}{19}}{\binom{500}{20}}$. The numerator, I think, is the number of ways that a particular element could be in some group of 20. The denominator is the number of ways of selecting groups of 20 from 500. This comes out to 1/25.

Alternately, we could select some random element in the 500, and then find the probability of successively selecting elements to form a group of 20 and not select that element each time. That'd be $\displaystyle \frac{499}{500}\cdot \frac{498}{499}\cdot \cdot \cdot \frac{480}{481} = 24/25$. Taking 1-24/25 gives the same answer again.

Did I find an error in the book?