Problem Solving

Hi,

I got this question from the Magoosh Blog, and the answer to the question[Answer in the last] is Theoretically correct,
But I fail to understand where my reasoning has gone wrong

Magaoosh Blog Question Link
https://magoosh.com/gmat/2015/counting- ... -the-gmat/

Question :-

A shipping company has four empty trucks that will head out in the morning,
all four to the same destination. The clerk has four different boxes to ship to that same destination.
All four boxes could go on any one of the trucks,
or the boxes could be split up into any groupings and given to the trucks in any combinations (ie. two to one truck, one to another, and one to another).
In how many different ways could the boxes be put on the four trucks?

Below is my reasoning, Please point me where i am going wrong

Case 1 :- Grouping as one Box from 4 Boxes

B1, B2,B3,B4,B5 each has four ways to be shipped hence it is 4*4*4*4 = 256 ways


Case 2 :- Grouping as two Boxes from 4 Boxes it is 4c2 and it is 6 total groupings
Like below and each one in the group can be shipped in 4 ways

1) B1,B2 = (4*4) ways
2) B1,B3 = (4*4) ways
3) B1,B4 = (4*4) ways
4) B2,B3 = (4*4) ways
5) B2,B4 = (4*4) ways
6) B4,B6 = (4*4) ways

= 6(16)
= 64 ways

Case 3 :- Grouping as 3 boxes from 4 boxes it is 4c3 and and it is 4 total groupings
like below and each one in the group can be shipped in 4 ways

1) B1,B2,B3 = (4*4*4) ways
2) B2,B3,B4 = (4*4*4) ways
3) B1,B3,B4 = (4*4*4) ways
4) B1,B2,B4 = (4*4*4) ways

= 4(4*4*4)
= 256 ways

Case 4 :- grouping as 4 boxes from 4 Boxes it is 4c4 and it is 1 way
= 4 Ways

Total ways = Case 1 + Case 2 + Case 3 + Case 4

= 580 ways

But the Correct answer is 4*4*4*4 = 256 Ways
which is theoretically correct.