![]() And the reason we reduce it by how many ways the cards could be in order is because we don’t care about the order when it comes to combinations. ![]() So, what does it mean? What does this \(r!\) times everything in the denominator tell us? It tells us that we are adjusting our permutation formula by reducing it by how many ways the cards could be in order, to get our combination formula. The difference is that in our combination formula we have \(r!\) being multiplied by our \((n-r)!\), but we don’t have that in our permutation formula. So let’s talk about the only difference between the two, and what that difference is representing. Now, we have to be careful, because the formula for permutation and combination are very similar. Now r is the number of cards in the person’s hand at a time, so \(r=5\), and that, it tells us that right here, “dealt five cards.” Now, it doesn’t say it in our problem, but we are expected to know that there are 52 cards in a standard playing deck. So, since n is equal to our total number of playing cards, we know \(n=52\). Order doesn’t matter for this problem, so we know that we will need to use combination. So, for this problem we don’t care about the order in which the cards are placed in this person’s hand. How many different hands are possible for the poker player to have been dealt. Let’s take a look at an example of a combination problem.Ī person playing poker is dealt five cards. Now, each of these ways are equal, my goal in showing you each is just to help you get down to the foundation of where we got this permutation formula. Once we cross out our 8 through 1 here on the top and on the bottom, we’re left with \(12\times 11\times 10\times 9\). If we look closely, we can see that this \(12\times 11\times 10\times 9\), once we write it out in this way, when we think of it in terms of having 4 spots on a wall and 12 in the first, and when we take one away, we have 11, and so on, we can see that this is the exact same thing we have when we actually use our permutation formula. So, the total permutations, the different ways that we could arrange these 4 paintings, since we care about the order in which we place them on the wall, is \(12\times 11\times 10\times 9\). And once we choose one from that, we’re left with 9 to go in our Spot 4. Then once we choose one from our 11 options, we’re left with 10 to go on Spot 3. Now, once we’ve chosen one from our 12, we’re left with 11 to go in our second option, 11 different painting options. Now, in the first place on our wall there are 12 different painting options to choose from. You have four different places on a wall, so that’s 1, 2, 3, 4. And we’re left with \(12\cdot 11\cdot 10\cdot 9=11,880\).Īnother way to think through what is going on is this: Now, once we write it all out, we can see that our 8 through 1 on the bottom and the top can cancel out with one another. So, basically all we need to find is how many ways there are to do this. ![]() Well, if an art gallery wants to arrange them a specific way then the order must be important. How many ways are there to do this?Īlright, let’s look at our problem and identify whether or not order is important. We will look at the formula for both permutation and combination, as well as how to spot whether or not order is important.Īn art gallery has twelve paintings by a local artist and wants to arrange four of them on the same wall. They say, “a combination lock should really be a permutation lock.”Īlright, let’s take a look at a couple different problems. There is a little joke that people often make. Now, when order is not important, like in the first example, then it is a combination but, when order is important it is a permutation. If I tried 786, I would be denied access. However, if I tell you the password to my computer is 876, then the order of those numbers is important. So, in this case, order is not important. I could say I made a quiche with broccoli, bell peppers, and sweet potatoes, and it wouldn’t matter. When working with a problem where permutation or combination is needed, to distinguish which one, all you need to do is ask yourself the question “does order matter?”įor example, if I tell you that I made a quiche with sweet potatoes, broccoli, and bell peppers it doesn’t matter which order I say it in. Now, the first thing you need to know about permutation and combination is when to use them. ![]() ![]() In this video we’ll take a look at two different types of probability using permutation and combination. ![]()
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