Here is a diagram of some pies with different numbers of cuts,
(starting with zero):

(Click on the diagram to get a clean image, suitable for printing. Can
you cut the uncut pies to complete the diagram?)

In each pie:

How many cuts (lines) are there?

How many cut (line) crossings are there?

How many pieces (regions) are there?

Some things to consider:

If one cut crosses a second cut, then the second cut also crosses the
first cut. The count of all the crossings made by all cuts is always
an even number?

Is there a nice formula which relates the numbers of cuts, crossings,
and pieces?

In the pies shown here, all the cut crossings are in the interior of the
pie. How to count a crossing where cuts cross at the edge of the pie?

In the pies shown here, all the cut crossings involve only two cuts.
How to count crossings where more than two cuts cross at the same point?
(Is there a way to save your original formula?)

For a particular number of cuts, in how many different ways can the
same number of pieces be cut?

To get the maximum number of pieces, each cut should cross all the other
cuts. (Is that true?) Is that always possible?