How many ways can I sort 50 distinct items into 2 lists with no repetition, order matters? I must use all $50$ items, but either list can be empty.
I know that the default answer is $2^k$ for $k$ elements when order does not matter. However, I am not sure how to arrive at the answer when order does matter. (As an aside: if there were three lists, would it be $3^k$ if order did not matter?)
I have attempted to break down the problem in the title as $3$ items into $2$ lists. The total number of combinations is $24$, but I am still unsure of how to extrapolate that number into a more general form. I have looked into a "stars and bars" solution but that doesn't give me $24$ either. 
The answer appears to be $4!$ for $3$ items, but I have only vague ideas as to why. Appealing to the multiplication principle with repetitions forbidden, I proposed that we have $4$ boxes, where the first box can be either the first, second, third, or none of the items. Is this the correct way to think about it? If so, is $51!$ the answer when there are $50$ items?

How many ways can you arrange 51 objects?  The 50 items, plus a divider that splits the items into two lists?
