I am trying to calculate how many randomly generated ids I need to produce for there to be a 1% probability I get a duplicate id.
There are $2^{128}$ possible ids.
I understand this is just the birthday problem with a big number:
$0.01=1-\frac{\operatorname{nPr}\left(2^{128},x\right)}{2^{128x}}$
However, I don't know how to solve this algebraically. I cannot plug in values for x and compute the probability because $2^{128}!$ is too large to handle.
Can you show me how to solve this algebraically, or get the numbers small enough to compute? If neither is possible, then how can I get an estimate?