This is a continuation of previous post:

How many distinct numbers can be formed by using four 2s and the four arithmetic operations .

For example:

(note that some binary operations do not make sense without parenthesis)

I have no idea about how to approach this problem (since I am not very comfortable with combinatorics). So any help will be appreciated.

**Edit[29 May 2017]:** This problem has been solved in the comments below.** **