3D Sum using Indirect Function

Hi Excel gurus.

I want to make spreadsheet that will sum different sheets. But the number and names of these sheets should change dependant on dropdown list.

I can use 3d sum such as =SUM('1:4'!A1:A15) (1 and 4 are sheet names) also I can use Indirect function such as SUM(INDIRECT("'"&A6&"'!a1:a15"))

But what I want is combination of both of these functions. I wanna sum particular ranges in different sheets which user will have a chance change its names. Used that function but getting REF error. SUM(INDIRECT("'"&A2&":"&B2&"'!a1:a15"))

Illustrated my problem in excel file and tried to to as clear as I could be.

I appreciate any help coming from you.

Thanx in advance