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Combinations vs. Permutations - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

In Algebra II, sometimes we will be asked to solve systems of equations three variables. When solving these systems of equations, a 3D coordinate system is necessary since systems of equations with three variables are not linear. Therefore, solving these systems of equations by graphing is not possible. Solving by substitution would be difficult, so we often solve by addition and elimination.

Well common problem that students have is to distinguish the difference between permutations and combinations. They're both distinguishing you know you have a certain number of things and you're taking a certain number from that but how we know when to use each can be a source of confusion. So in general the sort of rule of thumb that I use is that with permutations order matters okay and so what I mean by that is you pick one thing you pick another, a permutation is going to allow a different result if you switch the order of those 2 okay. A combination will not, you choose 2 things, it doesn't matter what the order is going to be. So permutation order matters, combination order doesn't okay.
Another way you can think of it is permutation is a way that things are organized sort of in a linear fashion. You go first, second, third, fourth, fifth so on and so forth, so you sort of place them in a spot. Whereas combinations are sort of just a collection of objects so you put a bunch of things into a bag and the order in that bag doesn't really matter. So you're sort of dealing with a linear versus the bag or order matters versus it doesn't. Either way permutation is going to be a line order matters, combination is going to be a bag order doesn't okay. So either way you think of it as long as you know that order matters for a permutation you're okay. So what I want to do, is just do a fairly simple example and start to distinguish the difference between the 2. So we have a running race there're 10 people on it and there's 2 things we're concerned with, the first is how many different ways are there to medal? So that takes into account gold, silver, bronze or otherwise first second, third place. The other one is how many different groupings could there be on the medal stand? So basically there you have a trio of people no matter what that trio is, it's going to be the same. So if the gold and the bronze person switch places they're still the same combination on that medal stand.
Okay so we said that order doesn't matter for choosing okay so order doesn't matter means that we are dealing with this sort of general grouping okay. So this is going to be a choose, order doesn't matter first and third can switch is the same combination of everything. And in this case we're dealing with 10 people so this is just going to be 10 choose 3. The difference with that and permutation is that with permutation order matters. So if we switch the first and the third person we're going to get a completely different permutation, we've switched the gold and the bronze we're going to get a completely different result. So this one the different ways are that a medal is a permutation. The numbers are exactly the same so this is still 10 and 3 but instead of choosing we end up using a permutation okay. So basically choosing order doesn't matter, permutation order does pick a way to remember that and stick with it up till now I don't really have any good sort of memory devices but hopefully you can come up with one.
The way I do is just order doesn't matter for choosing and I go with that, but somehow just get it into your head which is which.

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