The slices of cake are different (think about an actual cake, even if you try to cut exactly the same, some slices will have different frosting, be slightly bigger/smaller, etc.).

The problem with an answer of $6^{10}$ is that each person is choosing "a", meaning one, slice of cake. Therefore the first person has $10$ choices, the second $9$, etc, giving answer choice (C). If you wanted to do people choosing slices of cake, you'd first need to choose which $6$ of the $10$ slices are used (so order doesn't matter), and then give out one slice to each person (be careful here, as then this is with no repetitions). This would give $\binom{10}{6} \cdot 6!$ which also simplifies to (C).