I am going to work with the idea that all the cookies are somehow different from each other and all the chips are as well
There are 4 places to put the first chip
4 for the second
4 for the third and 4 for the forth.
That is 4^4 = 256 possible combinations.
Now how many of those have one cookie with three and one cookie with one and 2 cookies with none?
There are 4 ways to chose 3 chips frm the four, ill melt those together so they cannot seperate.
Now there are 4 posible cookies to put that melted mess into.
Now there are 3 possible cookies left for the other one.
So that is 4*4*3 = 48 possibilies
How many ways can all 4 chips can go into one cookie. That is easy .. 4 ways.
So there are 48+4 = 52 ways out of 256 ways that 3 or more chips are in any one cookie.
So There must be 256-52 = 204 ways to distribute the chips so that there is NOT more than 2 chips in any one cookie.
ie 204/256 = 51/64
This answer is the same as your given one so I guess it is right.
NOTE: I had had not been trying to reproduce the given answer I most likely would have come up with some other solution.
To quote myself:
With probabillity it is easy to see why the correct answer is correct but often (it seems ) impossible to see why the wrong answer is wrong! That is why it is so frustrating!
I think that the reason 35 is wrong is that your method assumes that the 4 stars all all distinct and also assumes that the bars are not.
In this question neither are distinguishable
You can work it out like i have by assume all are distinct or you can probably do it some other way where neither are distinct but you can not go half way like you have done.