Question

Write regular expression for the set of strings of 0's and 1's with at most one pair of consecutive 1's.

Answer 1

This should be the DFA for the given language. Since not mentioned explicitly, have taken $\epsilon$ as a member too.

We can frame the equation by looking at the Automata
$\epsilon$ + (0 + 1)0$^*$ + (0 + 1)10$^*$ 

or you can simplify it to 
$\epsilon$ + 00$^*$ + 10$^*$  +  010$^*$ + 110$^*$ 
$\epsilon$ + 0(0$^*$ + 10$^*$) + 1($\epsilon$ + 1)0$^*$