# Regular expression for language with even number of 0's and 1's [duplicate]

Let $\Sigma=\{0,1\}$. What is the regular expression for the language of all strings with an even number of $0$'s and an even number of $1$'s?

If we only require an even number of $0$'s, the language $(1^*)\mid(1^*01^*01^*)^*$ works. But once there is a requirement on both $0$'s and $1$'s, I'm not sure how to do it.

But if you prefer, it can be done directly, in two steps. Start by making two regular expressions $\alpha$ and $\beta$.
• First $\alpha$ specifies strings with even number of $1$'s, and exactly two $0$'s, one of which is the last symbol of the string.
• Similarly $\beta$ specifies strings with odd number of $1$'s, and exactly two $0$'s, one of which is the last symbol of the string.
Then build an expression for strings that have an even number of $0$ ánd an even number of $1$'s based on $\alpha$ and $\beta$.