Hey Zach,
Do you mean how do you know which to use? If so, basically, you use the permutation formula if order is a relevant concern (i.e. if changing the order would result in a legitimately different scenario), and you use the combination formula if all you care about is what/who you wind up with, and not how those things are arranged. Some examples:
How many different poker hands are possible? --> Combination, because all you care about is what cards are in your hand... arranging them in a different order does not create a different situation.
How many different ways can the competitors place in the race --> Permutation, because if A gets 1st place and B gets 2nd and C gets 3rd, that's a different scenario from if A gets 1st and C gets 2nd and B gets 3rd.
How many different ways can I arrange the vases in a row on the mantle? --> A permutation
I'm packing four shirts for a trip. I own 15 shirts. How many different combinations of four shirts to bring are possible? --> Combination, because they'll all get jumbled together in my suitcase. Order is irrelevant.
I've packed four shirts for a trip. In how many different orders can I wear them? --> Permutation, since, by my specifications, I want to count Day 1 Stripes, Day 2 Plaid, Day 3 Red, Day 4 Polka-dots as a different scenario from Day 1 Stripes, Day 2 Plaid, Day 3 Polka-dots, Day 4 Red.
But I may have misunderstood your question and you not have actually been asking how to distinguish between the two. If that's the case, my apologies!
Ashley Newman-Owens
GMAT Instructor
Veritas Prep
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