Do the values of the two * inside the ( ) need to remain unchanged for every repetition of ( )?
For example, 110011001100 is part of this language set, but 1100100010 isn't?
Or can the values of the two * change for every repetition of the ( )?
Do the values of the two * inside the ( ) need to remain unchanged for every repetition of ( )?
For example, 110011001100 is part of this language set, but 1100100010 isn't?
Or can the values of the two * change for every repetition of the ( )?
For any two words $w_1$ and $w_2$, the regular expression $\texttt{((}w_1\texttt{)*(}w_2\texttt{)*)*}$ is equivalent to $\texttt{(}w_1\texttt{|}w_2\texttt{)*}$. Hence, both your examples are elements of the language generated by $\texttt{(1*0*)*}$.
In this setting, it is impossible to specify $w_1$ and $w_2$ occur equally often with a regular expression (except for a finite number of possibilities). Otherwise, you would be able to generate a non-regular language, in this case $\{ 1^n 0^n \mid n \in \mathbb{N} \}^\ast$.
There is no memory in regular expressions. If $R$ is a regular expression, then $R^*$ means "any sequence of strings, each of which matches $R$". So, in your example of $(1^*0^*)^*$, the outer star means "any sequence of strings, each of which matches $1^*0^*$. $110$, $1$, $000$ and $11$ all match $1^*0^*$, so $110\,1\,000\,11$ matches $(1^*0^*)^*$.
Indeed, since $1$, $0$ and $\varepsilon$ all match $1^*0^*$, every binary string matches $(1^*0^*)^*$.