Regarding bars and stars method in Marbles question

#1

In the hint of marbles question, there is a concept star and bars method.

There is another approach that came into my mind before watching the hint.

firstly, we take k marbles, (as 1 marble per category)

Now, If we take k categories as spaces. And we are left with
(n-k) marbles.

Then, we can say that, each of (n-k) marbles will have k options.

So, according to the above method, we have (n-k)^(k) ways/options.

But according to the above logic, my answer is getting wrong.

Plz tell me, where my approach is wrong.

#2

this also considers order of selection of marbles thats why answer in larger than expected. To be more clear lets assume we are left with three marbles and k=2, so your formula considers these two cases as different while both are same!

  1. marble-1,2 of type 1 and marble-3 of type 2.
  2. marble -1,3 of type 1 and marble-2 of type 2.
    here at the end you have 2 marbles of type 1 and 1 marble of type-2 hence both are equivalent!
#3

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