A strange algebra math question that many failed|algebra simplification

We are familiar with the factorial, but this is the double factorial. This is fundamental. And this is subfactorial. So these are types of factorial. So to explain what we have on the board, we are first of all going to start from known to unknown. So let's talk about factorial to start with Now remember that if I say four factorial, this simply means four times three times two. times one Right, and if I say three factorial is three times two times one So you literally have to multiply from the starting number until you reach one—the product of everything you get. It's going to be your factorial, but what is a double factorial? A double factorial simply means that it now depends on if we have a An even number or an odd number. All right Now seven is an odd number, so seven double factorial—let's begin with an even number, for example. If you have six double factorial, it simply means the product of all even numbers from six down to two. Product of all even numbers from six to one That should be six times four times two times one, which is six double factorial. Now, if we have an odd number in this case, which is seven They should be the product of all odd numbers from seven to one or from one to seven. Okay, so that brings us to this, which is seven double factorial. Is equal to seven times five times three times one The product of all odd numbers from seven down to one. So I'm going to leave it like this. Yes. Now, let's go to this hashtag No, it's not really a hashtag; it's called primordial now. Primordial simply means the product of all um What's it called again? All prime numbers from whatever number you're given up to one Okay. Now for example, if I say seven Primordial is going to be the product of all odd Numbers from seven to one, so it's going to be seven times five times three times two. Okay, remember that one is not A prime number, you know, okay, now in this case we have five. Prime, so it's going to be the product of all prime numbers from five down to one. So that simply means that five Primordial is equal to five times three times two. So we are going to leave this at this point Now to the main point—this is where I want you to pay attention. This is subfactorial. It has a different meaning from factorial. But before that, have you clicked the subscription button? Have you? If not, then this is the right time to do it, because you might forget. Yes. So if you're actually enjoying this and this is the first time you've seen us, please Hit the subscription button and turn on your notification bell so you don't miss our amazing videos. We upload our videos every day, daily. We upload two to three times at your convenience. Please tune in and check what we have for you. Okay, and then give us a thumbs up if you are actually enjoying this video Thank you very much. So back to what I was saying. This is where I want you to pay attention. So attention, please. Alright, let's continue; now we have five subfactorials. Before you understand what a subfactorial is. Then you need to understand what a factorial is. Sorry about the background noise Okay.FONT 妖精の森の奥深く、古い石の門の前には、小さな穴が開いていた。その穴から、微かに光が漏れてきていた。 Three factorial means how many ways you can arrange three objects? All right How many ways can you arrange three objects in a straight line? Three factorial means three times two times one. Times one which is six, but if you want to do it manually Anyway, this is still done manually, but without a formula—if you want to do that. That is going to be, for example, if you are giving A, B, and C to arrange. In how many different ways can we arrange it? We already have the first arrangement. We can still keep a constant and swap the positions of c and b. All right. Now we're done with it. Let's pick b. Then we have a c, right? We can still write b and interchange the positions of a and c. So we have c; now we can decide to keep c constant. Alright, C, now first of all, when we write C, let's bring A and B together, right? Then we can decide to switch in the next part. So you can see that you have six ways to arrange three objects in a straight line. Now what is a subfactorial? A subfactorial is how many times? Can you arrange a particular number of objects? Alright, in how many ways can you arrange them such that none of the objects remains in its original position? The original position of each of those objects should not be repeated in any one now. This is what I mean. We have a b c. Remember that a stays in the first position, right? B is the second position, C is the third position, as in when referring to the original. or the main arrangements Okay, now let's go. Anyone who remains in their original position will be removed. That is when we are looking for a subfactorial. So look at this now you have a b c because a is retaining its position. Here, this is not about thinking; it is about returning to this position, which is the first position. This is not going to happen now when we get here, B is not coming back to this position. A is not retaining this position, but C is third and still returning to the third position here. So no, we are not taking this. So we go to this; b is not returning its position. C is not returning to this position. A is not returning to his position. So this is a goal for us. A is not returning to its position; A is not returning to position B. not returning its position, so this is it. Now let's move on to this: C is not Return it to this position, but be returning it to its original position. This is Second composition, so this is not going well, from what we have discussed or from what we've just discussed, you can see that 3 factorial equals 6, not 2. Is there a formula for this now? 36 factorial is setting us three factorial. x times 1 over 0 factorial minus 1 over 1 factorial, okay, then plus 1 over 1 factorial minus 1 over 2 factorial, sorry, minus 1 over 3 factorial, so I'm going to stop at 3 factorial because this is 3 sub factorial, so if we solve this we have 3 factorial multiplied by 1 over now 0 factorial Yes, I have a video I made on this topic, which is why 0 factorial equals 1. I gave three reasons why 0 factorial equals 1, so I'm going to attach that video here, okay. So you can go and check it out after watching this video. Alright, so we have 1 over 1 minus 1. over 1 plus 1 over 2 because 2 factorial is 2, right, minus 1 over 6, that is 3 factorial, so we 3 factorial multiplied by this gives us 0 already, so what we have is 1 over 2 minus 1. Over 6 now, to make them have the same denominator, we multiply 2 by 3 and multiply 1 by 3. by 3 as well, so it's going to be times 3 times 3, so we have 1 times 3 is 1 divided by 2 times 3 times 6 minus 1 times 3 is 3, sorry, divided by 6 minus 1 over 6, alright? So we have 3 factorial. is equal to 3 minus 1, which is 2, over 6. Now remember that 3 factorial is the same thing as 6, so this and This equals we have 2, so we can see how we arrived at 2 as our answer. Now let us solve the one we have. On the board, okay, so according to the question, we are looking for 5 subfactorial, okay, so 5 sub factorial is going to be 5 factorial multiplied by multiplied by 1 over 0 factorial minus 1 over 1 factorial okay plus 1 over 2 factorial minus 1 over 3 factorial plus 1 over 4 factorial minus 1 over 5 factorial; now remember that we need to stop at 5 because we are looking for 5 subfactorial Okay, so this is equal to 5 factorial multiplied by then this is 1 over 1, right, minus 1 over 1. plus 1 over 2 factorial is 2 minus 1 over 3 factorial is 6 plus 1 over 4 factorial is 4 times 3 times 2 times 1, so 4 times 3 is 12 times 2 is 24, so we have 24 minus now 1 over 5 factorial 5 factorial is 5 times the entire quantity of this, so it's the same as 5 times 24, which will give US 120 right 5 times 4 is 20, right? 0 parry 2, 5 times 2 is 10, plus 2 is 12, so it's definitely Going to give us 120, all right, so we have 1 over 120, perfect, so we have 5 factorial multiplied By remembering this, we get 0, so we are left with the entire expression: 1 over 2 minus 1 over 6. plus 1 over 24 minus 1 over 120, so this is equal to 5 factorial multiplied by now we need the least common multiple To be 120, sorry, all denominators to be 120, right? So 2 times what gives us 120? That is 60, 60. Also multiply 1 by 60: 1 times 60 is 60. 60 over 2 times 60 is 120. Minus now 125 by 6, we get 20, right? Yeah, so 1 times 20 over 6 times 20, so we're going to have 20 over 120 plus 1 for this 24. It's going to be 5, so 1 times 5 is 5, 24 times 5 is 120, then minus 1 over 120, so we have 5 factorial. multiplied by everything will now be 60 minus 20 plus 5 minus 1, everything divided by 120, so Let's continue from here, all right? We are almost there, we are almost done. So this is what we have. We have 5 subfactorial equal to 5 factorial multiplied by 16 minus 20, which is 40, 40 plus 5 is 45. 45 minus 1 is 44, so we have 44 over 120. So we have 5 sub factorial; remember that 5 factorial Is 120 right multiplied by 44 over 120? Now, this we divide; we have one remaining, so 5 sub. factorial is equal to 44, so that is what we are going to substitute for 5 sub factorial now let Let's go back and substitute everything we got. Remember that 7 double factorial is 7 times 5. times 3 times 1, so let us solve this: we have 7 times 5 times 3 times 1 divided by now 5 sub 5 Primordial gave us 5 times 3 times 2, so 5 times 3 times 2, now multiplied by 44, which is our 5. sub factorial now if we do this, this we divide itself to give us 1/3; we divide itself to give us 1 so what we have is 7 over 2 multiplied by 44 so 2 we divide this to give us 22 so what we have is 22 multiplied by 7; now, 7 times 2 is 14, carry 1; 7 times 2 is 14 plus 1 is 15, so you can See that the answer is 1 5 4, interesting, right? Thank you so much for watching. Don't forget to Watch the video where I said the reasons why zero factorial is equal to one. Thank you so much for watching.