Determining which set is larger

Solution 1:

"Then, this is a set of 500 elements where each element is repeated five times, so there are $$ \frac{500!}{100\cdot 5!} $$ distinguishable permutations of this set."

No, There are $$ \frac{500!}{(5!)^{100}} $$ distinguishable permutations of this set. This is because there are $5!$ ways for the ones to be permuted and then for each of those there are $5!$ ways for the twos to be permuted so $(5!)^2$ ways for the ones and twos to be permuted and this repeats until we get the answer above. This is less than $10^{1000}$ so it works (By a python script it has 927 digits)