Understanding Permutations and Combinations

In this video, we're going to focus on permutations and combinations. But what exactly is a permutation? And how is it different from a combination? A permutation is associated with arranging things in different order. Combinations, you're simply concerned about combining things. For a permutation, the order matters, but for a combination, the order does not matter. I guess the best way to explain this is with an example. Let's say if we have three letters, A, B, C. We could arrange them in this order, A, B, C, or we could say C, A, B. Now, even though we have the same three letters, the order is different. So, these are two different permutations. The number of permutations is two. However, the number of combinations is one. These two are considered to be the same in terms of a combination. But in terms of a permutation, they're considered to be two separate things. So just make sure you understand that a permutation, the order matters, the way you arrange it matters, and for a combination, the order doesn't matter. You just want to combine things. So if you have ABC and CAB, you still have the same three letters combined in a group. So for a combination, they're the same. Let's use another example to illustrate this. Let's say if we have four letters, A, B, C, and D. And let's choose two of the four letters. In how many different ways can we arrange two of the four letters? And also, how many different ways can we combine two of the four letters? Make a list and use that to determine the number of permutations and combinations of choosing 2 out of the 4. And then we'll talk about how to use an equation to get that same answer. So we can choose AB, we can choose AC, or we can choose AD. We can choose BA, BC, BD. We're only using each letter once every time we select two out of the four. We can also use CA, CB. and CD and also DA, DB, DC. So notice that there's a total of 12 different ways we can arrange 2 out of the 4 letters. So the number of permutations in this example is equal to 12. Now what about the combinations? Now if you recall, for a combination, the order doesn't matter. So take a look at AB and BA. In terms of permutations, these are counted as two separate things. But for a combination, you're combining the same letters. And since the order doesn't matter for a combination, they're counted once as a combination. So if we're going to count AB, we can't count BA. If we're going to count, let's say, AC, we can't take into account CA. If we're going to count AD, we have to eliminate DA. If we're going to use BC, we need to get rid of CB. If we're going to use BD, we can't use DB. And finally, if we're going to use CD, we've got to get rid of DC. So notice that the number of combinations is equal to 6. 1, 2, 3, 4, 5, 6. So now you can clearly see the difference between a permutation and a combination. So just remember, a permutation, the order matters, and for a combination, the order does not matter. Now how can we calculate these answers? Is there an easier way in which we can find the value, as opposed to making a list of all the different possibilities? The first equation you need to know is NPR. This helps you calculate the permutations. Now, there's four letters and we're choosing two, so it's going to be 4P2. We're choosing two out of a group of four. Now, the equation NPR is equal to N factorial divided by N minus R factorial. In this case, we can see that N... is 4 and r is 2. So n minus r that's going to be 4 minus 2. Now 4 minus 2 is 2. So we have 4 factorial divided by 2 factorial. So what exactly is 4 factorial? 4 factorial is 4 times 3 times 2 times 1. You start with this number, and you multiply 4 by every integer all the way to 1. 2 factorial is simply 2 times 1. So, we can cancel 2 times 1, and we're left with 4 times 3, and we know that 4 times 3 is equal to 12, which is what we have here. Now, how can we calculate the combination? What is the formula that we can use? nCr is equal to n factorial divided by n minus r factorial divided by r factorial. So basically, in terms of a permutation, a combination is equal to nPr divided by r factorial. This portion right here is nPr, and then divided by r. So we have 4, C2 in this example. So N is 4, and R is 2. So this is equivalent to 4 factorial divided by 2 factorial times 2 factorial. And we know that 4 factorial is 4 times 3 times 2 times 1. 2 factorial is 2 times 1. And we have another 2 factorial. it turns out that 2 times 2 is 4 so we can cancel those two with the four on top we can ignore one because one times anything won't change the value. So what we have left over is 3 times 2. 3 times 2 is equal to 6. And so now you understand how to use the equation, and also you understand how to make a list to determine the number of permutations and combinations. So now let's work on some example problems. In how many different ways can you arrange three books on a shelf from a group of seven? Now go ahead and try this problem. Pause the video, take a minute, and feel free to work on it, and then unpause it to see the solution. So is it a permutation or a combination? Does the order matter? Whenever you see the keyword arrange, typically it's a permutation. The order is important. So what we need to write is 7P3. We're choosing three books from a group or from a total of seven. So this is going to be 7 factorial divided by the difference between 7 and 3. 7 factorial is 7 times 6 times 5 times 4 times 3 times 2 times 1. 7 minus 3 is 4, and 4 factorial is 4 times 3 times 2 times 1. one. So we can cancel these numbers leaving behind 7 times 6 times 5. Now 6 times 5 is 30 and we know that 3 times 7 is 21 so 30 times 7 is 210. This is the answer. In how many different ways can we arrange five books on a shelf? How is this problem different from the last problem? And is it still a permutation? Well, we're still trying to arrange books so the order matters. It's still a permutation. But you can use the fundamental counting principle to get the answer. So we want to arrange five books on the shelf, right? So there's five positions to place the five books. In the first position, we can choose any of those five books. So we have five options. Now, once we place the first book, in the first position, there's four books left over to choose from. So we can put any of the four books in the second position. Now that we've placed two books, we have three left over. So we can put any of those three books in the third position. Now we have two books left over, so we can put any of those two in the second to last position. And in the last position, we can only put the last book there. So it's going to be 5 times 4 times 3 times 2 times 1. Thank you. And that's another way in which you can solve these problems. 5 times 4 is 20. 3 times 2 is 6. If 2 times 6 is 12, 20 times 6 is 120. Now, in terms of a permutation, here's how you can calculate it. First, you need to find out what is the total number of books. In this problem, we only have one number. The total number of books is 5. And we're choosing all 5 books from a group of 5, so it's going to be 5p5. Using the formula NPR is equal to N factorial over N minus R factorial. N is 5, but R is 5 as well. So this is going to be 5 factorial over 0 factorial. Now, this is not undefined. 0 factorial does not equal 0. 0 factorial equals 1. Make sure you know that. That's just something to know. If you're wondering why, that's just the way it is. I don't have an answer for you. So, this is going to be 5 factorial over 1, and we know that 5 factorial is 5 times 4 times 3 times 2 times 1, and this is equal to 120. How many teams of four can be produced from a pool of 12 engineers? So is this a permutation or is it a combination? What do you think? Does the order matter? So let's say if I select four individuals, John, Sue, Sally, and Chris. Does it really matter if I select John, Chris, Sally, Sue? It's the same team of four. So in this problem, the order doesn't matter. Therefore, it's a combination. So we're choosing 4 from a total of 12. So it's going to be 12c4. And this is equal to n factorial, or 12 factorial, divided by n minus r factorial, that's 12 minus 4 factorial, times r factorial, or 4 factorial. So 12 minus 4 is equal to 8. Now, if you don't want to write 12 times 11 times 10 times 9 times 8 times 7 all the way to 1, here's what you can do. Notice that you have an 8 factorial on the bottom. So you want to write 12 just before you get to 8. So 12 factorial is 12 times 11 times 10 times 9 times 8 factorial. Because 8 factorial will go from 8 to 12. to 1. You don't need to write all of it. Stop at a factorial because we can cancel it in the next step. Now 4 factorial, I'm going to write that out. That's 4 times 3 times 2 times 1. So let's cancel a factorial. That's going to save us some writing space. 4 times 3 is 12, so we can get rid of these two. Then 10 divided by 2 equates to 5. So what we now have is 11 times 5 times 9. 11 times 9 is 99. Now what is 99 times 5? Well, if you want to do that without your calculator, think of it this way. 99 is 100 minus 1. Let's distribute. 100 times 5 is 500. 5 times 1 is 5. 500 minus 5 is 495. And so we can choose 495 teams of 4. from a pool of 12 engineers. And so that's it for this video. Thanks for watching. If you want to find more videos that I've created in Algebra, Treg, PreCal, Calculus, Chemistry, and Physics, just visit my channel. And you can find my playlists on those topics. Well, I changed my mind. I just realized that there are some other problems that I need to go over that's related to this topic. How many different ways can you arrange the letters in the word Alabama? This is a very common question that you might see in this type of topic. And here's what you need to do. First, count the number of letters that are in the word Alabama. There's a total of seven letters. So it's going to be seven factorial on the top of the... fraction and on the bottom divide by the letters that repeat there's only one letter that repeats and it's a and a repeats four times so we're going to divide it by four factorial so therefore this is going to be seven times six times five times four factorial divided by four factorial So we could cancel these two we know that 6 times 5 is 30 and 7 times 30 is 210 So that's how many different ways you can arrange the letters in the word, Alabama. Let's try another example. What about the word? Mississippi in class I've seen this a lot so it's a very common example, so I'm going to use it So first let's count how many letters that we have There's a total of 11 letters, so it's going to be 11 factorial on top, divided by... Now, let's find the letters that repeat. I repeats 4 times, so we're going to divide it by 4 factorial. S repeats 4 times, so another 4 factorial. And p repeats twice, so 2 factorial. So this is going to be 11 times 10 times 9 times 8 times 7 times 6 times 5 times 4 factorial. divided by I'm going to leave the first four factorial the same I'm not going to change it and the other form going to write it as 4 times 3 times 2 times 1 and Then 2 factorial 2 times 1 so we can cancel 4 factorial and Let's see what else can we cancel? Well, we know that uh 3 times 2 is Equal to 6 so we can cancel those and also 4 times 2 is equal to 8, so we can cancel that as well. So what we have left over is 11 times 10 times 9 times 7 times 5. So 11 times 10 is 110, and 7 times 5 is 35. Now, 110 times 9, I believe that's 990. Now we need to multiply 990 by 35. And I'm going to use a calculator at this point. So this will give you 34,650. So that's the answer. And now that's it for this video. That's all I got, so thanks for watching, and have a great day.

For a permutation, the order matters, but for a combination, the order does not matter. I guess the best way to explain this is with an example. Let's say if we have three letters, A, B, C. We could arrange them in this order, A, B, C, or we could say C, A, B. Now, even though we have the same three letters, the order is different.

So, these are two different permutations. The number of permutations is two. However, the number of combinations is one. These two are considered to be the same in terms of a combination.

But in terms of a permutation, they're considered to be two separate things. So just make sure you understand that a permutation, the order matters, the way you arrange it matters, and for a combination, the order doesn't matter. You just want to combine things.

So if you have ABC and CAB, you still have the same three letters combined in a group. So for a combination, they're the same. Let's use another example to illustrate this. Let's say if we have four letters, A, B, C, and D. And let's choose two of the four letters.

In how many different ways can we arrange two of the four letters? And also, how many different ways can we combine two of the four letters? Make a list and use that to determine the number of permutations and combinations of choosing 2 out of the 4. And then we'll talk about how to use an equation to get that same answer.

So we can choose AB, we can choose AC, or we can choose AD. We can choose BA, BC, BD. We're only using each letter once every time we select two out of the four. We can also use CA, CB. and CD and also DA, DB, DC.

So notice that there's a total of 12 different ways we can arrange 2 out of the 4 letters. So the number of permutations in this example is equal to 12. Now what about the combinations? Now if you recall, for a combination, the order doesn't matter.

So take a look at AB and BA. In terms of permutations, these are counted as two separate things. But for a combination, you're combining the same letters.

And since the order doesn't matter for a combination, they're counted once as a combination. So if we're going to count AB, we can't count BA. If we're going to count, let's say, AC, we can't take into account CA.

If we're going to count AD, we have to eliminate DA. If we're going to use BC, we need to get rid of CB. If we're going to use BD, we can't use DB.

And finally, if we're going to use CD, we've got to get rid of DC. So notice that the number of combinations is equal to 6. 1, 2, 3, 4, 5, 6. So now you can clearly see the difference between a permutation and a combination. So just remember, a permutation, the order matters, and for a combination, the order does not matter. Now how can we calculate these answers?

Is there an easier way in which we can find the value, as opposed to making a list of all the different possibilities? The first equation you need to know is NPR. This helps you calculate the permutations.

Now, there's four letters and we're choosing two, so it's going to be 4P2. We're choosing two out of a group of four. Now, the equation NPR is equal to N factorial divided by N minus R factorial.

In this case, we can see that N... is 4 and r is 2. So n minus r that's going to be 4 minus 2. Now 4 minus 2 is 2. So we have 4 factorial divided by 2 factorial. So what exactly is 4 factorial?

4 factorial is 4 times 3 times 2 times 1. You start with this number, and you multiply 4 by every integer all the way to 1. 2 factorial is simply 2 times 1. So, we can cancel 2 times 1, and we're left with 4 times 3, and we know that 4 times 3 is equal to 12, which is what we have here. Now, how can we calculate the combination? What is the formula that we can use? nCr is equal to n factorial divided by n minus r factorial divided by r factorial. So basically, in terms of a permutation, a combination is equal to nPr divided by r factorial.

This portion right here is nPr, and then divided by r. So we have 4, C2 in this example. So N is 4, and R is 2. So this is equivalent to 4 factorial divided by 2 factorial times 2 factorial.

And we know that 4 factorial is 4 times 3 times 2 times 1. 2 factorial is 2 times 1. And we have another 2 factorial. it turns out that 2 times 2 is 4 so we can cancel those two with the four on top we can ignore one because one times anything won't change the value. So what we have left over is 3 times 2. 3 times 2 is equal to 6. And so now you understand how to use the equation, and also you understand how to make a list to determine the number of permutations and combinations. So now let's work on some example problems.

In how many different ways can you arrange three books on a shelf from a group of seven? Now go ahead and try this problem. Pause the video, take a minute, and feel free to work on it, and then unpause it to see the solution. So is it a permutation or a combination?

Does the order matter? Whenever you see the keyword arrange, typically it's a permutation. The order is important.

So what we need to write is 7P3. We're choosing three books from a group or from a total of seven. So this is going to be 7 factorial divided by the difference between 7 and 3. 7 factorial is 7 times 6 times 5 times 4 times 3 times 2 times 1. 7 minus 3 is 4, and 4 factorial is 4 times 3 times 2 times 1. one. So we can cancel these numbers leaving behind 7 times 6 times 5. Now 6 times 5 is 30 and we know that 3 times 7 is 21 so 30 times 7 is 210. This is the answer. In how many different ways can we arrange five books on a shelf?

How is this problem different from the last problem? And is it still a permutation? Well, we're still trying to arrange books so the order matters.

It's still a permutation. But you can use the fundamental counting principle to get the answer. So we want to arrange five books on the shelf, right? So there's five positions to place the five books.

In the first position, we can choose any of those five books. So we have five options. Now, once we place the first book, in the first position, there's four books left over to choose from. So we can put any of the four books in the second position. Now that we've placed two books, we have three left over.

So we can put any of those three books in the third position. Now we have two books left over, so we can put any of those two in the second to last position. And in the last position, we can only put the last book there. So it's going to be 5 times 4 times 3 times 2 times 1. Thank you. And that's another way in which you can solve these problems.

5 times 4 is 20. 3 times 2 is 6. If 2 times 6 is 12, 20 times 6 is 120. Now, in terms of a permutation, here's how you can calculate it. First, you need to find out what is the total number of books. In this problem, we only have one number.

The total number of books is 5. And we're choosing all 5 books from a group of 5, so it's going to be 5p5. Using the formula NPR is equal to N factorial over N minus R factorial. N is 5, but R is 5 as well. So this is going to be 5 factorial over 0 factorial. Now, this is not undefined.

0 factorial does not equal 0. 0 factorial equals 1. Make sure you know that. That's just something to know. If you're wondering why, that's just the way it is. I don't have an answer for you.

So, this is going to be 5 factorial over 1, and we know that 5 factorial is 5 times 4 times 3 times 2 times 1, and this is equal to 120. How many teams of four can be produced from a pool of 12 engineers? So is this a permutation or is it a combination? What do you think? Does the order matter?

So let's say if I select four individuals, John, Sue, Sally, and Chris. Does it really matter if I select John, Chris, Sally, Sue? It's the same team of four.

So in this problem, the order doesn't matter. Therefore, it's a combination. So we're choosing 4 from a total of 12. So it's going to be 12c4.

And this is equal to n factorial, or 12 factorial, divided by n minus r factorial, that's 12 minus 4 factorial, times r factorial, or 4 factorial. So 12 minus 4 is equal to 8. Now, if you don't want to write 12 times 11 times 10 times 9 times 8 times 7 all the way to 1, here's what you can do. Notice that you have an 8 factorial on the bottom.

So you want to write 12 just before you get to 8. So 12 factorial is 12 times 11 times 10 times 9 times 8 factorial. Because 8 factorial will go from 8 to 12. to 1. You don't need to write all of it. Stop at a factorial because we can cancel it in the next step.

Now 4 factorial, I'm going to write that out. That's 4 times 3 times 2 times 1. So let's cancel a factorial. That's going to save us some writing space.

4 times 3 is 12, so we can get rid of these two. Then 10 divided by 2 equates to 5. So what we now have is 11 times 5 times 9. 11 times 9 is 99. Now what is 99 times 5? Well, if you want to do that without your calculator, think of it this way.

99 is 100 minus 1. Let's distribute. 100 times 5 is 500. 5 times 1 is 5. 500 minus 5 is 495. And so we can choose 495 teams of 4. from a pool of 12 engineers. And so that's it for this video.

Thanks for watching. If you want to find more videos that I've created in Algebra, Treg, PreCal, Calculus, Chemistry, and Physics, just visit my channel. And you can find my playlists on those topics. Well, I changed my mind.

I just realized that there are some other problems that I need to go over that's related to this topic. How many different ways can you arrange the letters in the word Alabama? This is a very common question that you might see in this type of topic.

And here's what you need to do. First, count the number of letters that are in the word Alabama. There's a total of seven letters. So it's going to be seven factorial on the top of the... fraction and on the bottom divide by the letters that repeat there's only one letter that repeats and it's a and a repeats four times so we're going to divide it by four factorial so therefore this is going to be seven times six times five times four factorial divided by four factorial So we could cancel these two we know that 6 times 5 is 30 and 7 times 30 is 210 So that's how many different ways you can arrange the letters in the word, Alabama.

Let's try another example. What about the word? Mississippi in class I've seen this a lot so it's a very common example, so I'm going to use it So first let's count how many letters that we have There's a total of 11 letters, so it's going to be 11 factorial on top, divided by...

Now, let's find the letters that repeat. I repeats 4 times, so we're going to divide it by 4 factorial. S repeats 4 times, so another 4 factorial.

And p repeats twice, so 2 factorial. So this is going to be 11 times 10 times 9 times 8 times 7 times 6 times 5 times 4 factorial. divided by I'm going to leave the first four factorial the same I'm not going to change it and the other form going to write it as 4 times 3 times 2 times 1 and Then 2 factorial 2 times 1 so we can cancel 4 factorial and Let's see what else can we cancel?

Well, we know that uh 3 times 2 is Equal to 6 so we can cancel those and also 4 times 2 is equal to 8, so we can cancel that as well. So what we have left over is 11 times 10 times 9 times 7 times 5. So 11 times 10 is 110, and 7 times 5 is 35. Now, 110 times 9, I believe that's 990. Now we need to multiply 990 by 35. And I'm going to use a calculator at this point. So this will give you 34,650.

So that's the answer. And now that's it for this video. That's all I got, so thanks for watching, and have a great day.

Transcript for:Understanding Permutations and Combinations

Transcript for:
Understanding Permutations and Combinations