How do I calculate the number of permutations of a specific binary string?

Question

For a binary string of 6 elements with three ones and three zeros in each string, how do I calculate the total number of string permutations? For example, 3 possible strings would be 100011, 111000 and 101010.

Answer 1

The number of binary strings with $m$ zeros and $n$ ones, is $\binom{n+m}{n}=\binom{n+m}{m}$. There are $n+m$ "positions," and you can choose $m$ of them to place the zeros; the ones will fill the rest.