in how many ways 5 different chocolates be distributed . . .

We must distribute 5 chocolates among 4 children. We know that any child can get any number of chocolates. This means that 3 children could get 1 chocolate and 1 child could get 2 chocolates; or 1 child could get all 5 chocolates and 3 children could get 0 chocolates; or 1 children could get 3 chocolates, 2 children could get 1 chocolate, and 1 child could get 0 chocolates; etc. etc.

This is pretty complex, but we can actually think about it in a pretty simple way. Since which child gets which chocolate next doesn't depend at all on which children already have chocolates, we can think of the distribution of each piece of chocolate independently.

The first chocolate can go to any of the 4 children. That's 4 options.

For each of those options, the second chocolate can go to any of the 4 children. That's another 4 options. This means that there are 4*4=16 options for how the first two chocolates can be given away.

For each of those options, the third chocolate can go to any of the 4 children. That's another 4 options. This means that there are 4*4*4=64 options for how the first three chocolates can be given away.

For each of those options, the fourth chocolate can go to any of the 4 children. That's another 4 options. This means that there are 4*4*4*4=256 options for how the first four chocolates can be given away.

For each of those options, the fifth chocolate can go to any of the 4 children. That's another 4 options. This means that there are 4*4*4*4*4=1024 options for how all four chocolates can be given away. This gives us the correct answer, E.