How to prove that for any words $w_1, ..., w_n$ of alphabet $\{0,1\}$ regular expression $w_1^*w_2^*...w_n^*$ doesn't represent language $\{0,1\}^*$?

Question

How to prove that for any words $w_1, ..., w_n$ on alphabet $\{0,1\}$ the regular expression $w_1^*w_2^*...w_n^*$ doesn't represent language $\{0,1\}^*$?

Answer 1

Suppose there exist such $w_1, ..., w_n$ (without loss of generality all words have length more than $0$). There exist such $k$, that $2^k>k^n$. There are $2^k$ words of length $k$ in language $\{0,1\}$ and not more than $k^n$ words of length $k$ in language represented by given regular expression.