How can everyone have unique iris and fingerprints? After a certain amount of human beings have lived on earth, wouldn't it be possible to exhaust all possible combinations?
The same principle applies to DNA. Lets say counting all the nucleotides on a strand of DNA we find that there are a total of $N$. We know there are 4 types of nucleotides, thus the total number of possible DNA sequences equals $4^N$. This is exponential, and since $N$ is immensely huge, $4^N$ will be unthinkably huge as well. However, since $4^N$ is a finite number, how can DNA be guaranteed to be unique? Doesn't that contradict the pigeon hole principle?