Probability of choosing same set of unique objects over time without replacement I have $1024$ unique objects, and each round I start with all of them and choose $7$ uniformly at random without replacement. If I do this $n$ times, what is the probability distribution that I will have a collision (pick the same $7$ as I have in a previous round)?
If this can be generalized to $N$ unique objects choose $K$, it would be useful to compare choices of parameters.
I vaguely think I could start answering this question but I'm far from an expert in probability and would greatly appreciate some help.

The number of different choices of $7$ of the $1024$ objects is $\binom{1024}7$. This is a very large number; call it $C$. 
If $n=2$, there is a collision with probability $\frac 1C$ and so no collision with probability $\frac{C-1}C$. If $n=3$, to avoid a collision we first have to choose differently in the first two rounds (probability $\frac{C-1}C$), then avoid both of those combinations in the third round (probability $\frac{C-2}C$), and so the probability of no collision is $\frac{(C-1)(C-2)}{C^2}$. Similarly the probability of no collisions in $n$ rounds is $$\frac{(C-1)(C-2)\cdots(C-n+1)}{C^{n-1}}.$$
This can be approximated as $\exp\big(-\frac{n(n-1)}{2C}\big)$. See this wikipedia page for more details.
