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“This video. We re going to focus on permutations and combinations. But what exactly is is a permutation and how is it different from a combination our permutation is associated arranging things in different order combinations. You simply concern about combining things for a permutation the order matters.
But for 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 abc. We could arrange them in this order abc 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 tunes. However the number of combinations is 1 these two are considered to be the same in terms of a combination or in terms of a permutation.
They re considered to be two separate things so just make sure you understand that 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 a b c.
And c. A b. 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 and 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 two out of the four and then we ll talk about how to use an equation to get that same answer.
So we can choose a b. We can choose a c or we could choose a b. We can choose b a b c. Bd.
We re only using each letter. Once every time. We select two out of the four. We can also use ca c b.
And c d. And also da db. Dc. So notice that there s a total of twelve different ways.
We can arrange two out of the four 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 a b and b.
8. 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 a b. We can t count ba. If we re going to count let s say a c.
We can t take into account ca if we re going to count a d. We have to eliminate da. If we re going to use bc. We need to get rid of c d.
If we re going to use the bd. We can t use db. And finally if we re going to use c d. We got to get rid of dc.
So notice that the number of combinations is equal to six one two three four five six. So now you can clearly see the difference between a permutation and a combination so just remember a permutation the order of matters and the 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 needs to know is npr this helps you calculate the permutations now this for letters and which used into so it s going to be for p 2. Which is in 2 out of a group of 4.
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 q. 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 nc r. Is equal to n factorial divided by n minus r. Factorial. Divided by our sector.
So basically in terms of a permutation a combination is equal to np r. Divided by r factorial. This portion right here is np r. And then divided by r.
So we have 4c 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 q times q is 4. So we can cancel those twos with the foreign song we d ignore 1 because 1 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 and how many different ways can you arrange 3 books on the shelf from a group of 7. 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 combination does the order matter. Whenever you see the key word.
Arrange typically hit the presentation. The order is important so we ll need to write is 7 p. 3. We choose in 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. 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 and how many different ways can we arrange 5 books on the shelf. How is this problem different from the last problem and is it still a permutation well we re still trying to arrange book so the order matters it s still a presentation but you can use the fundamental counting principle to get the answer so we want to arrange 5 books on the shelf right so there s 5 position to place the 5 books in the first position.
We can choose any of those 5 books so we have 5 options now let s can place the first book in the first position. There s 4 books left over to choose from so we can put any of the 4 books in the second position now that we ve placed two books. We have to be left over so we could put any of those three books in the third position. Now we have 2 books left over so we can put any of those two in a second for 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.
And that s another way in which you could 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 which used in all 5 books from a group of 5.
So it s going to be 5 p. 5. Using the formula n p. R.
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 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 4 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 automatic so let s say if i select for individuals john s to sally and crisps does it really matter. If i select john kriss sally sue. It s the same team of 4.
So in this problem the order doesn t matter therefore. It s a combination so we ll choose in for from a total of 12. So it s going to be 12 c. 4.
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 from 1. Is 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 a factorial will go from 8 to 1. And you don t need to write all of it stop at a factorial because we can cancel it in the next step now for factory. 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 2. And then 10 divided by 2 equates to 5.
So 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 five hundred 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 an algebra trade precal calculus chemistry and physics just visit my channel and you can find my playlist 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 his we need to do first count the number of letters that are in the word alabama. There s a total of 7 letters.
So it s going to be 7 factorial on the top of the fraction and on the bottom divided by the letters that we paint. There s only one letter that repeats and it s a and a repeats four times. So we re going to divide it by 4 factorial. So therefore.
This is going to be 7 times 6 times. 5 times. 4. Factorial divided by 4 factorial.
And 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 examples.
I m going to use it so first let s count how many letters that we have there s a total of eleven letters. So it s going to be 11 factorial on top divided by now let s find the letters that repeat i repeat four times. So we re going to divide it by 4 factorial s. Repeat four times.
So another four factorial and pe 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 factory of the same. I m not going to change.
It and the other forms 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 a 3 times. 2 is equal to 6. So we can cancel those and also 4 times 2 is equal to 8. So we 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 a 990 now we need to multiply 990 by 35 and i m going to use the calculator at this point. So this will give you thirty four thousand six hundred and fifty. 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 ” .
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This video tutorial focuses on permutations and combinations. It contains a few word problems including one associated with the fundamental counting principle. Permutations are useful to determine the different number of ways to arrange something where as combinations is useful for determining how many ways to combine something when the order does not matter such as selecting members to form a committee. In a permutation, the order matters. Examples include repeated symbols or arranging letters in a word such as alabama or mississippi. This video also discusses the basics of permutations and combinations using letters such as ABCD.
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