Need help calculating an amount of combinations for my job

Ok, so consider the following, the items are labeled (1) to (31) and each is either on the list or it is not. So we have 2 different states for each of these items. Now the question is, how many possible combinations exist.

Hint 1: >!If there exist x possibilities for item (1) and y possibilities for item (2), then there exists x\*y possibilities for the list (1)(2).!<

Hint 2: >!As a\*a\*a\*a = a\^4!<

Solution: >!2\^31 though you might want to subtract 1 if the possibility of an empty list does not exist.!<
You start with the total number of permutations of the entire list in any case >!31!!<

Then you need to factor out the number of ways to order things in the "picked" and not picked" subgroups (because only picked / not picked matters, you don't care about the order in which anything is listed).  >!For the case where the combination has 10 items, that's 10! ways of arranging the "picked" items and 21! ways of arranging the list items that were "not picked"  ((31!)/((10!)(21!)))!<

Then, since you can have a combination with [1-31] items, you need to sum up the number of combinations of each allowable length to get the total number of valid combinations. >!31 choose 1 + 31 choose 2 + ... + 31 choose 31!<

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