Let L = { w in (0 + 1)* | w has an even number of 1s }. That is, L is the set…

Correct regular expression: 0*(10*10*)*

The initial 0* allows any number of leading zeros, including pure-zero strings and the empty string.

Each block 10*10* contains exactly two 1s. Repeating this block any number of times therefore always adds 1s in pairs. So every generated string has an even number of 1s.

Conversely, any binary string with an even number of 1s can be read as some leading zeros followed by pairs of 1s, with any number of zeros between and after the two 1s in each pair.

Therefore, 0*(10*10*)* represents exactly the language L.