![]() ![]() P(10,3) = 720.ĭon’t memorize the formulas, understand why they work. Permutation: Listing your 3 favorite desserts, in order, from a menu of 10. Permutation: Picking a President, VP and Waterboy from a group of 10. Here’s a few examples of combinations (order doesn’t matter) from permutations (order matters).Ĭombination: Picking a team of 3 people from a group of 10. Writing this out, we get our combination formula, or the number of ways to combine k items from a set of n: Which means “Find all the ways to pick k people from n, and divide by the k! variants”. In our case, we get 336 permutations (from above), and we divide by the 6 redundancies for each permutation and get 336/6 = 56. If we want to figure out how many combinations we have, we just create all the permutations and divide by all the redundancies. So, if we have 3 tin cans to give away, there are 3! or 6 variations for every choice we pick. If you have N people and you want to know how many arrangements there are for all of them, it’s just N factorial or N! Wait a minute… this is looking a bit like a permutation! You tricked me! So we have $3 * 2 * 1$ ways to re-arrange 3 people. Well, we have 3 choices for the first person, 2 for the second, and only 1 for the last. For a moment, let’s just figure out how many ways we can rearrange 3 people. This raises an interesting point - we’ve got some redundancies here. Either way, they’re equally disappointed. If I give a can to Alice, Bob and then Charlie, it’s the same as giving to Charlie, Alice and then Bob. ![]() Well, in this case, the order we pick people doesn’t matter. How many ways can I give 3 tin cans to 8 people? In fact, I can only afford empty tin cans. Let’s say I’m a cheapskate and can’t afford separate Gold, Silver and Bronze medals. If we have n items total and want to pick k in a certain order, we get:Īnd this is the fancy permutation formula: You have n items and want to find the number of ways k items can be ordered:Ĭombinations are easy going. Where 8!/(8-3)! is just a fancy way of saying “Use the first 3 numbers of 8!”. What’s another name for this? 5 factorial!Īnd why did we use the number 5? Because it was left over after we picked 3 medals from 8. This is where permutations get cool: notice how we want to get rid of $5 * 4 * 3 * 2 * 1$. Unfortunately, that does too much! We only want $8 * 7 * 6$. ![]() To do this, we started with all options (8) then took them away one at a time (7, then 6) until we ran out of medals. The total number of options was $8 * 7 * 6 = 336$. We picked certain people to win, but the details don’t matter: we had 8 choices at first, then 7, then 6. Silver medal: 7 choices: B C D E F G H. ![]() Gold medal: 8 choices: A B C D E F G H (Clever how I made the names match up with letters, eh?).We’re going to use permutations since the order we hand out these medals matters. How many ways can we award a 1st, 2nd and 3rd place prize among eight contestants? (Gold / Silver / Bronze) We’re using the fancy-pants term “permutation”, so we’re going to care about every last detail, including the order of each item. Let’s start with permutations, or all possible ways of doing something. ![]()
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