What is Complementary counting?

Example 10.2 (Colliding identifiers)

Thirty client order identifiers are drawn uniformly and independently from 10 000 values. The chance that all differ is $\prod_{k=0}^{29}(1 - k/10\,000) \approx 0.957$, so at least two coincide with probability about 0.043. The approximation $1 - e^{-n(n-1)/(2N)}$ gives the same to three decimals, and it shows the scale: collisions become likely once $n$ is of the order of $\sqrt N$.