It is also convienient to extend this definition to negative by if , . By fiddling a little, we can see that for , we have:

- For and , so divides the product of any consecutive integers.
- Putting in the binomial theorem gives -- so the number of subsets of an -set is . There are many proofs of this fact. An easy one is by induction on . Write for the total number of subsets of an -set. Then and for , . (Pick a point in the -set and observe that there are subsets not containing it and subsets containing it.
- -- so in any finite set the number of subsets of even sizes equals the number of subsets of odd sizes.

as unless or | ||