Permutations And Combinations
Transcript
Welcome to the video on Permutations & Combinations. Permutations and combinations are very important because you will encounter questions in both and that will be from this topic. Also, you have to be proficient with permutations and combinations so as to be able to answer the questions related to probability. So basically, if you have permutations and combinations as a strong point you can actually name two topics, permutations and combinations, and probability.
Because in probability also there is a lot of permutations and combinations involved. So I will have many questions for this topic and be very complicated with this. In this video, we'll be learning about the principle of counting and combinations with all different objects. So let's begin with principle of counting. So the first principle of counting is the multiplication principle.
So this principle says that if an operation is performed in m different ways, another operation can be performed in n different ways. Then the two operation in succession can be performed in m into n ways. So there's a first operation that can be performed with m different ways. This word different is really important. And there is another operation that can be performed in n different ways, then they can perform in succession, in m into n ways.
So this is the multiplication principle. So basically if you want to perform operation 1 and operation 2, then the total number ways of doing so is m multiplied by n. I just give you an example. Suppose a hall has 10 doors, and you want to enter the hall, and exit the hall by different gates.
So in how many ways can you do so? So the number of ways of entering the halls will be 10, right, why? Because there are 10 doors, you can enter through any other door. So there are 10 ways of entering the hall, which are through 10 different doors. Once you have entered the hall, you will not be coming back from the same door. Because in the question it is given that we have to exit through a different gate.
So once you have actually entered the hall, that entry gate, you will discard that one. Now you have remaining 9 doors from which you can exit, and these are different doors. So the total number of ways of entering the hall and exiting the hall by using different gates is 10 x 9 which equals to 90.
There is one more principle of counting known as addition principle. So this principle says that if two operations are independent from each other, okay, this independent- Is very important. If two operations are independent from each other, and the first operation can performed in m ways, and the second operation can be performed in n ways, then either of the two operations can be performed in m + n ways.
So essentially what is happening is that there is an operation 1 that can be performed in m ways, operation 2 that can be performed in n ways, and you want to perform either operation 1 or operation 2. So you are not basically performing the two operations in succession or simultaneously. You have to perform either of the two operations.
So in this case, if the operations are independent from each other, then the number of ways of doing so will be equal to m + n. So I'll just give an example. Suppose in a class, there are 20 students in total. Out of which there are 8 girls and 12 boys. Now there's a teacher who wants to select either a girl or a boy for some activity to be performed.
The teacher does not want to select the girls and the boys together, it just 1 girl or 1 boy. So in how many ways can the teacher do so? So number of ways of selecting girls is 8. Number of ways of selecting boys is 12. And either girls are selected or boys are selected, they are not selected in succession or simultaneously.
So there will be a plus sign and hence, the answer is 20, in this case. Now, let's move on to combinations. And here we will be considering that all objects are different. So basically, if some of the objects are the same, so in that case, the number of ways of selecting objects will be slightly different. So we are not going into that in this video.
For this video, we'll be considering that all the objects are different, and you have to select some objects from all these different objects. So the number of ways of selecting r things, From n different things. So this different is very important. At a time is denoted by nCr which equals to factorial n upon factorial r, factorial n minus r.
So essentially if you have total of n different things, and you have to select r things from these n different things, then the number of ways of doing so is equals to nCr. So for example, say you have a 5 different chocolates, And you want to choose any 2 of these 5 different chocolates. So number of ways of doing so is, 5C2 which equals to factorial 5 upon factorial 2 and factorial 3, which is equals to 10.
Now pause this video and verify this thing, that actually the number of ways of choosing 2 chocolates from 5 different chocolates is equals to 10. Now, let's see some special cases. Suppose, again we have n different objects and you have to select r objects from these n different objects, such that 8 things are always included. So, I'll write it down.
N different objects, select r different objects, such that k things are always included. So pause this video and try to find the expression for this one. Now let's see what is the expression for this thing actually.
So the expression will be n- k C r- k, right? Now, I explained how this expression actually came up. So basically, we had included k things, right? So that k things that are included, we already know that these are the things, right? So essentially, we have to now choose r- k things.
And hence the r- k appeared here right? Also there were a total of n different objects right? k of them were already included, so the remaining objects are n- k, right? So out of these n-k, we have to now select r-k objects. Hence n-k C r-k. Now, suppose we have n different objects and we have to select r objects such that k things are always excluded.
So the expression for this one will be n-k Cr, why? Because they were total n different things, right? And you have decided to exclude k things so the remaining number of things is n-k. And from these n-k things, you have to select r things hence the expression for the number of ways of selecting r things from n things such that k things are always excluded is n-k Cr.
Now suppose we have a case in which we have to select r things from n total things such that k things are always not together in any selection. So essentially, what we have to do is that we have k objects and these k things should not be together in any selection.
You have to still select r things from n things but these k things should not be together in a selection. So it can be that some of the k things is present, some of the k things are not present, but all the k things should not be present together in any selection. So the number of ways of doing so is total number of ways of selecting r objects from n objects minus the number of ways in which these k things are always included.
So that is essentially n-k C r-k. So I hope you are getting what I did here. nCr is the total number of ways of selecting r objects from a total of n different objects. And n-k C r-k is the number of ways of selecting r objects out of n different objects such that k things are always included.
But we want that these k things should not be together, should not be always included, right? So hence we have to subtract n-k Cr-k from nCr, so as to obtain the expression of the number of ways of selecting r objects from n different objects such that k things are not together.
Read full transcriptBecause in probability also there is a lot of permutations and combinations involved. So I will have many questions for this topic and be very complicated with this. In this video, we'll be learning about the principle of counting and combinations with all different objects. So let's begin with principle of counting. So the first principle of counting is the multiplication principle.
So this principle says that if an operation is performed in m different ways, another operation can be performed in n different ways. Then the two operation in succession can be performed in m into n ways. So there's a first operation that can be performed with m different ways. This word different is really important. And there is another operation that can be performed in n different ways, then they can perform in succession, in m into n ways.
So this is the multiplication principle. So basically if you want to perform operation 1 and operation 2, then the total number ways of doing so is m multiplied by n. I just give you an example. Suppose a hall has 10 doors, and you want to enter the hall, and exit the hall by different gates.
So in how many ways can you do so? So the number of ways of entering the halls will be 10, right, why? Because there are 10 doors, you can enter through any other door. So there are 10 ways of entering the hall, which are through 10 different doors. Once you have entered the hall, you will not be coming back from the same door. Because in the question it is given that we have to exit through a different gate.
So once you have actually entered the hall, that entry gate, you will discard that one. Now you have remaining 9 doors from which you can exit, and these are different doors. So the total number of ways of entering the hall and exiting the hall by using different gates is 10 x 9 which equals to 90.
There is one more principle of counting known as addition principle. So this principle says that if two operations are independent from each other, okay, this independent- Is very important. If two operations are independent from each other, and the first operation can performed in m ways, and the second operation can be performed in n ways, then either of the two operations can be performed in m + n ways.
So essentially what is happening is that there is an operation 1 that can be performed in m ways, operation 2 that can be performed in n ways, and you want to perform either operation 1 or operation 2. So you are not basically performing the two operations in succession or simultaneously. You have to perform either of the two operations.
So in this case, if the operations are independent from each other, then the number of ways of doing so will be equal to m + n. So I'll just give an example. Suppose in a class, there are 20 students in total. Out of which there are 8 girls and 12 boys. Now there's a teacher who wants to select either a girl or a boy for some activity to be performed.
The teacher does not want to select the girls and the boys together, it just 1 girl or 1 boy. So in how many ways can the teacher do so? So number of ways of selecting girls is 8. Number of ways of selecting boys is 12. And either girls are selected or boys are selected, they are not selected in succession or simultaneously.
So there will be a plus sign and hence, the answer is 20, in this case. Now, let's move on to combinations. And here we will be considering that all objects are different. So basically, if some of the objects are the same, so in that case, the number of ways of selecting objects will be slightly different. So we are not going into that in this video.
For this video, we'll be considering that all the objects are different, and you have to select some objects from all these different objects. So the number of ways of selecting r things, From n different things. So this different is very important. At a time is denoted by nCr which equals to factorial n upon factorial r, factorial n minus r.
So essentially if you have total of n different things, and you have to select r things from these n different things, then the number of ways of doing so is equals to nCr. So for example, say you have a 5 different chocolates, And you want to choose any 2 of these 5 different chocolates. So number of ways of doing so is, 5C2 which equals to factorial 5 upon factorial 2 and factorial 3, which is equals to 10.
Now pause this video and verify this thing, that actually the number of ways of choosing 2 chocolates from 5 different chocolates is equals to 10. Now, let's see some special cases. Suppose, again we have n different objects and you have to select r objects from these n different objects, such that 8 things are always included. So, I'll write it down.
N different objects, select r different objects, such that k things are always included. So pause this video and try to find the expression for this one. Now let's see what is the expression for this thing actually.
So the expression will be n- k C r- k, right? Now, I explained how this expression actually came up. So basically, we had included k things, right? So that k things that are included, we already know that these are the things, right? So essentially, we have to now choose r- k things.
And hence the r- k appeared here right? Also there were a total of n different objects right? k of them were already included, so the remaining objects are n- k, right? So out of these n-k, we have to now select r-k objects. Hence n-k C r-k. Now, suppose we have n different objects and we have to select r objects such that k things are always excluded.
So the expression for this one will be n-k Cr, why? Because they were total n different things, right? And you have decided to exclude k things so the remaining number of things is n-k. And from these n-k things, you have to select r things hence the expression for the number of ways of selecting r things from n things such that k things are always excluded is n-k Cr.
Now suppose we have a case in which we have to select r things from n total things such that k things are always not together in any selection. So essentially, what we have to do is that we have k objects and these k things should not be together in any selection.
You have to still select r things from n things but these k things should not be together in a selection. So it can be that some of the k things is present, some of the k things are not present, but all the k things should not be present together in any selection. So the number of ways of doing so is total number of ways of selecting r objects from n objects minus the number of ways in which these k things are always included.
So that is essentially n-k C r-k. So I hope you are getting what I did here. nCr is the total number of ways of selecting r objects from a total of n different objects. And n-k C r-k is the number of ways of selecting r objects out of n different objects such that k things are always included.
But we want that these k things should not be together, should not be always included, right? So hence we have to subtract n-k Cr-k from nCr, so as to obtain the expression of the number of ways of selecting r objects from n different objects such that k things are not together.
