00:01
Once again, welcome to new problem.
00:04
When you think about organizing elements in a set, we can always think about permutations, and we could also think about combinations.
00:21
So we can think about permutations and combinations.
00:25
When it comes to permutations, the variations, the variations, the variations in which a set of elements in variations in which a set of elements can be organized.
01:01
So this is permutations and there's something special about permutations and the arrangements and the arrangements the arrangements of elements in a permutations are such that the order of arrangement is the order of arrangement is significant so the way you organize for example if i had abc this would be different from bac and it would also be different from aac and it would also be different from a and so these are permutations, different arrangements that i can organize elements.
01:59
And this is opposed to combinations.
02:03
So in combinations, you don't have that problem.
02:07
So in combinations, in combinations order doesn't make a difference.
02:18
So the order doesn't make a difference when it comes to combinations.
02:23
For example, all these three would just represent the same combination.
02:28
So we don't have any distinction between these elements.
02:32
In terms of formulas for combinations and permutations, when it comes to permutations, or rather when it comes to combinations, ncr would be n factorial all of n minus r factorial and then times r factorial.
02:52
When it comes to permutations, it's just npr and this is the same as n factorial all of n minus r factorial.
03:03
Remember, 4 factorial will be the same as 4 times 3 times 2 times 1.
03:08
0 factorial is always 1.
03:11
So that's the definition of factorials.
03:18
That's the definition of factorials.
03:22
In statistics, so we have a new problem in statistics and statistics, the standard terminology, there is a standard terminology that we're looking at, it's called homo, homo, stacicity.
03:50
Homoscadasticity.
03:51
So this terminology, we wanna determine we wanna determine how many different ways, how many different ways.
04:04
This terminology can be organized.
04:12
So we wanna determine how many different different ways this terminology can actually be organized.
04:19
So the first thing we're going to do is we already understand that we're dealing with permutations here.
04:34
We're dealing with permutations and the fact that we're dealing with permutations is we're going to have n and 1 and 2 and r.
04:47
So these are just different ways that we can organize the computation.
04:53
So n1 factorial and 2 factorial and r factorial.
04:59
And this happens if some items are identical.
05:07
So if some items are identical, we're going to run the problem like that.
05:12
So the number of ways, the number of ways n objects can be grouped into our different classes.
05:35
So n objects can be grouped into our different classes with n sub i.
05:45
Being the size of the ith class.
05:53
So n objects, the number of different ways that n objects can be grouped into different classes.
06:01
And so this is the first class, this is the second class, this is the third class.
06:06
So those are different ways.
06:08
And so we do know that n factorial is pretty much n times n minus 1 times n minus 2 all the way up until times 1, which is the last one.
06:28
And we did give an example right here, how the factorial plays out.
06:32
So this would be n, this would be n minus 1, this would be n minus 2.
06:38
And then, of course, this would be the last one, which is n minus 3, and n minus 3...