Calculate every component separately:
0.12 = 0.1*0.1 = 0.01
0.13 = 0.1*0.1*0.1 = 0.01*0.1 = 0.001
and sum them all up:
0.1 + 0.12 + 0.13 = 0.1 + 0.01 + 0.001 = 0.111
There are 3 different components. The term 0.1 has 1 digit after the decimal point. The term 0.12 has 2*1 = 2 digits after the decimal point. The term 0.13 has 3*1 = 3 digits after the decimal point. The maximum number of digits after the decimal point is 3. So, the answer has exactly 3 digits after the decimal point. Look through the answer options and choose one with 3 digits after the decimal point.
If there are some properties of the answer that can be identified directly from the problem and only one answer option fits them, it MUST be the right answer.
For example, in the problem we figured out that the answer has to have exactly 3 digits after the decimal point and only one answer option (0.111) fit the property. So, it's the right answer.