How many ways are there of placing 6 marbles in 4 bowls, if any number

It wasn't stated explicitly, but we all assumed in our solutions, that all the marbles are distinct/different (think of different colors or numbered marbles). Then the above solutions are correct. The number of possibilities to place 6 distinct/different marbles in 4 bowls is \(4^6.\)

If the marbles are all identical, the bowls are distinct, then what is different between the distributions is the particular number of marbles in each bowl.
In this case, the above formula you mentioned should be used. For example, 6 identical marbles can be placed in 4 bowls in (6 + 4 - 1)C(4 - 1) = 9C3 = 84 ways. In the original question, since none of the listed answers is 84, the hidden assumption was that the marbles are non-identical, which I think it should have been stated explicitly.

For \(n\) identical marbles and \(r\) bowls, a way to justify the formula is as follows: think of the of the marbles placed in slots instead of bowls. The slots are aligned, created such that there are \(r-1\) dividing internal walls, something like this: [o|ooo|...| |o| ], where [ and ] represent the two outer walls of the slots. In the first slot there is one ball, in the second three balls,..., there is an empty slot, just one ball, and the last one is also an empty slot.
In each slot, we can place any number of marbles between 0 and r.
Imagine that we have \(n+r-1\) places, because we have \(n\) marbles and \(r-1\) dividing walls, and we just have to decide in this string of length \(n+r-1\) where to place the walls (or equivalently, where to place the marbles). This can be done in (n + r - 1)C(r - 1) different ways, or equivalently, (n + r - 1)Cn.