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Need help calculating an amount of combinations for my job

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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.!<
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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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