Here's the complete solution, just for fun.
If you give the seven rabbits these names:
a b c d e f g
Then the seven click patterns are:
1: a b 2: a b c 3: b c e 4: d e f 5: c d e 6: d f g 7: f g
If you make a list of the patterns that activate each rabbit, you get this:
a: 1 2 b: 1 2 3 c: 2 3 5 d: 4 5 6 e: 3 4 5 f: 4 6 7 g: 6 7
You can see that rabbits "a" and "g" are special: The only way to activate rabbit "a" is by using pattern 1 or 2. The only way to activate rabbit "g" is by using pattern 6 or 7. That narrows the search down to just four combinations--which is very doable! You *could* try each of those four combinations in turn, and be done in a minute. OR:
You could notice that both patterns 6 and 7 will activate rabbits "f" and "g": The only difference is that pattern 6 will activate rabbit "d" as well. So, is it useful to activate rabbit "d"? No, not really. Pattern 5 activates the middle rabbits ("c", "d", and "e"). That leaves only the first two rabbits ("a" and "b"), and those two rabbits can be activated by pattern 1.
So you can see pretty much immediately that patterns 7, 5, 1 will solve the puzzle.
That's all you need to do! You don't actually need to try all four combinations--even though you could. (Considering all four combinations could lead you to another solution, or show that there aren't any.)