Hi Turkish,
This question isn't quite written up to the standards of the Official GMAT, but the "intent" of it is that we're meant to treat it as a Combinations question. In this way, the order of the 2 balls does NOT matter, meaning that if you pulled the "1 ball", then the "2 ball", then THAT result is the same as pulling the "2 ball" then the "1 ball", and we're not supposed to count that as two different events - it's one unique outcome. This means that we'll have to do one extra math 'step' at the end (we will have to divide the number of outcomes by 2).
Since the answer choices ARE numbers, and we're asked how many balls are in the bag, we can TEST THE ANSWERS.
We're told that there are balls numbers from 1 up to N in a bag. We're told that there are 55 ways to pull out 2 balls and NOT get the "3 ball." We're asked for the value of N.
Let's TEST Answer C first...
IF....there were 9 balls, then.....
there are 8 ways to NOT get the "3 ball" on the first grab
there are then 7 ways to NOT get the "3 ball" on the second grab.
(8)(7) = 56
However, we have to divide this by 2, since the "intent" of the question is that the order of the grabs does NOT matter.
56/2 = 28 total ways. This is TOO SMALL (it's supposed to be 55 ways). This means that we need MORE balls. Eliminate A, B and C.
Next, let's try Answer D:
IF....there were 12 balls, then....
there are 11 ways to NOT get the "3 ball" on the first grab
there are then 10 ways to NOT get the "3 ball" on the second grab.
(11)(10) = 110
However, we have to divide this by 2, since the "intent" of the question is that the order of the grabs does NOT matter.
110/2 = 5 total ways. This is a MATCH for what we were told, so there MUST be 12 balls in the bag.
Final Answer:
GMAT assassins aren't born, they're made,
Rich