My professor just said these are the genrating functions and then he went straight to examples. He never, not even once showed how these things actually generate the series. I tried plugging values for x and never got a reasonable output (obviously). He did not once say that you have to do polynomial division. So this entire topic just seemed like magic to me. Thank god you exist
Thank you so much. You explain things in a simple, easy to understand manner. Your videos are straight to the point so I'm not skimming through a week's worth of my lectures trying to understand this concept.
I SHOULD MAKE A STATUE IN YOUR HONOR! Tell me, how does it feel to save people for their impending doom in some random tests around the globe? -Brazilian computer science student
Wow, I wish my university taught like you did. Your simple use of real world problems abstracted mathematics was a very important (and missing) link that most of us don't get from a formal education.
at 12:40, you go from =0+1+2x+3x^2+4x^3... to (1,2,3,4,5,6,...) I'm not understanding this. How did you get (1,2,3,4,5,6,...)? Why doesn't it start with 0?
Thank you for the speedy response! I'd like to ask a follow up question about that. So, from: 0+1+2x+3x^2+4x^3... For our list of coefficients, we simply do not list 0. We obtain 1 from x^0. Then, we obtain 2 from the coefficient of x^1, obtain 3 from the coefficient of x^2, etc?
You do not list 0 because the first coefficient is the coefficient of x^0. 0x^0 + 1x^0 = 1x^0. If I had 2x^2 + 4x^2 then the list of coefficients would be (0, 0, 6, 0, 0, 0, ...)
@TheTrevTutor...at 9:47 shouldn't the first series have the formula (1-x^n)/(1-x) instead of (1-x^n+1)/(1-x) ....maybe theres a calculation error coz I rechecked using the G.P. summation formula
at 15:30 Why should we go higher then x^12 for x1??? since x1+x2+x3=12 and x1>=2 hence it should be x^2,x^3,x^4,x^5,x^6,x^7,x^8,x^9,x^10,x^11,x^12. Please clarify! Appreciate your response.
It's slightly simpler to write out functions that go on forever than cutting them off. It doesn't change the coefficients at all, so it's inconsequential if we go further or not.
Yo TrevTutor, I would love if you made another video on just the proofs. The longer proofs like (NOTp v q) ^ NOT(NOTq v r) v ( NOTp v r) being equivalent to a tautology tend to mess me up with how the laws work in them. Thanks mate.
There's no difference in the end, since the coefficients will still be the same even if we allow it to go higher. (This doesn't generalize to all questions)
Hi, Mr Trev, I have a question regarding the jellybean question.In the first conditon, it says even reds, but the second conditon says AT LEAST 14, does it give the restriction to the first condition? which in my assumption, its: 1 + x2 + x4 +x6. Corret me if I am wrong? thanks a lot
17:54 how to solve it further. I have watched complete series of generating functions mentioned on your channel. Is it the case of partial fraction decomposition and solve for each a0/(1-ax)^n ? Please reply m stuck.
Regarding long division of polynomials - aren't you suppose to line them up in decreasing order, i.e. -x + 1 instead of 1-x. How does this affect the answer? Also the 1/(1-x) solution is only true for |x| < 1. Are we assuming this?
So, if I understand correctly, there are 12 ways to solve the first question where x1+x2+x3=12 with 0≤xi≤6. You said that there are 3+4+5 ways of solving the equation, so that would be 12 ways in total, right?
thank you so much for your videos! Got a combinatorics puzzle. Could you please help me with it? Choose 6 numbers from 1 to 49 so that exactly two of them are consecutive. How many combinations?
I FOUND A MISTAKE !!!! in the question of red,blue and white shouldn't the red be (1+x^2+x^4+x^6) that's it because we have to give atleast 14 to the blue one,then how the red can go upto ...+x^20 expecting a reply soon thanks :)
It doesn't make a different. In the end we look at the coefficient of x^20. Letting red go up to x^20 won't affect the overall coefficient of x^20 after multiplying all the terms out.
In the jellybeans example ,isn't the generating function (1+x^2+x^4+x^6)(x^14+x^15+x^16+x^17+x^18+x^19+x^20)(1+x+x^2+x^3+x^4),since we have at least 14 blue jellys ?
you don't need to stop at x^6 in this first term because any number above 6 will never be chosen since we will need to add it to at least 14 in the second term
These are nothing but G.P. . Sum of infinte G.P = a/1-r, but for that we've to assume that |x| < 1. Can we do that , is it the legit way to do ? As x is a variable of no significance , can we assume its value and solve the G.P. ?
It is interesting function. I think the x range should have range. 1/(1-x) forms you explained in the long division but when we divide 1 by 1-x the remnant x must be smaller than 1-x therefore, the x range should fall x
How is that (1-x) goes 1 time into 1, x times into x and x² times into x², I'm not getting the meaning of any of these yet you say it like it's super obvious
He's just matching the power of the (1) in (1-x) with the highest power of the polynomial. When the number is 1, you subtract a 1-x from 1 to get x. Then you subtract x(1-x) from x to get x^2. You continue this to reach the infinite series 1+x+x^2+x^3.... The principle is similar to long division, which is why it is called polynomial long division.
![[Discrete Mathematics] Coefficient Extraction and Extended Binomial Theorem](http://i.ytimg.com/vi/ZyUb5UxBA9Q/mqdefault.jpg)
![[Discrete Mathematics] Coefficient Extraction and Extended Binomial Theorem](/img/tr.png)
Thank you, still helping students 5 years in the future.
6
at 16:20 why for the blue brackets. (x^2^ x^3^ x^4...) he takes x^2 at the first so he starts at x^2 (x^0 ^x^1 x^2^...) why he starts at x^0 ....
7
8
9
My professor just said these are the genrating functions and then he went straight to examples. He never, not even once showed how these things actually generate the series. I tried plugging values for x and never got a reasonable output (obviously). He did not once say that you have to do polynomial division. So this entire topic just seemed like magic to me. Thank god you exist
i hate how shitty college professors are man
Basically saved my life. That's the explaination I searched for months. A great video
Dude, you're saving my quarantined ass with this, my online lectures on this topic were very lackluster, thank you.
Dude you're a legend, after watching your videos I believe I can actually pass my exam!
I love how lucid yet simple this was.
I have a test tomorrow and I have just started. Glad to find this video. A useful resource.
In which class, you were at the time of writing this question?
I don't think generating functions are taught in schools
Thank you so much. You explain things in a simple, easy to understand manner. Your videos are straight to the point so I'm not skimming through a week's worth of my lectures trying to understand this concept.
I SHOULD MAKE A STATUE IN YOUR HONOR!
Tell me, how does it feel to save people for their impending doom in some random tests around the globe?
-Brazilian computer science student
hahaha so true greek computer science student
yes! - Azerbaijani computer science student
You nailed it on the head! Marine Corps, computer science student here!
Canada computer science student here.
Somehow it feels right to write here: Mathematics major, Turkey :)
Idk how but I feel like even the look of your words make things seem easier to understand. Better than my professor.
I hardly suq at english. Understood everything PERFECTLY. Subscription for being good at what you are doing!
Wow, I wish my university taught like you did. Your simple use of real world problems abstracted mathematics was a very important (and missing) link that most of us don't get from a formal education.
You are 100% correct,
Thanks very much stumbled on this video after 7-8. And this by far the best video to understand generating functions.
you have helped me beyond what i can explain :) thanks
Thank you so much. I am sending thanks from Istanbul Technical University. I even read the Brualdi's Combinatorics book but I understand well now.
This really helped me get my basics right. Thank You! Keep Uploading More.
woooohh what a life saver! crisp and clear explanation on generating functions
This is amazing. Good and appropriate examples.
I’m inspired to memorize these now!
I would recommend this to economics and statistics students big time
Hey TrevTutor, at 3:19, shouldn't you end the red jellybeans at x^6 only as it cannot possibly exceed that. Or maybe it doesnot matter??
You are saving my life at exam time here.
amazing teacher man
the first 5 minutes of this vidfeo gave me more understanding than a whole ass online lecture
THIS VIDEO IS AWESOME!!!!!
at 12:40, you go from
=0+1+2x+3x^2+4x^3...
to
(1,2,3,4,5,6,...)
I'm not understanding this. How did you get (1,2,3,4,5,6,...)? Why doesn't it start with 0?
0+1=1. The coefficient of x^0 is 1.
Thank you for the speedy response! I'd like to ask a follow up question about that.
So, from: 0+1+2x+3x^2+4x^3...
For our list of coefficients, we simply do not list 0. We obtain 1 from x^0. Then, we obtain 2 from the coefficient of x^1, obtain 3 from the coefficient of x^2, etc?
You do not list 0 because the first coefficient is the coefficient of x^0.
0x^0 + 1x^0 = 1x^0.
If I had 2x^2 + 4x^2 then the list of coefficients would be (0, 0, 6, 0, 0, 0, ...)
TheTrevTutor Thank you!!!!!
my heartiest thanx to you. This is superb and the best. Seriously you are the best teacher.
@TheTrevTutor...at 9:47 shouldn't the first series have the formula (1-x^n)/(1-x) instead of (1-x^n+1)/(1-x) ....maybe theres a calculation error coz I rechecked using the G.P. summation formula
Awesome Awesome Awesome...
I tried reading a book to learn this subject but my mind just got crushed out... So simple and amazing
so clear and easy to understand for me.
*clicks on video* this is exactly what I was looking for :)
YOU ARE A LIFE SAVER, Thank you man.
15:40 would you please explain to me why does this go to infinity?? and not to x¹²?? i couldnt understand the exolanation of the video
at 15:30 Why should we go higher then x^12 for x1??? since x1+x2+x3=12 and x1>=2 hence it should be x^2,x^3,x^4,x^5,x^6,x^7,x^8,x^9,x^10,x^11,x^12. Please clarify! Appreciate your response.
It's slightly simpler to write out functions that go on forever than cutting them off. It doesn't change the coefficients at all, so it's inconsequential if we go further or not.
Best on TH-cam I think.
4:05 how do i find the coeff of x^20
Wdym by sequence of x^n
But wouldnt you have to multiply the brakcets out before taking derivaitives
How we are going to get coefficient of x^12 at 2:20 ? Can you please elaborate? Appreciate :)
You made my day! man
Yo TrevTutor, I would love if you made another video on just the proofs. The longer proofs like
(NOTp v q) ^ NOT(NOTq v r) v ( NOTp v r) being equivalent to a tautology tend to mess me up with how the laws work in them. Thanks mate.
in the last question, shouldn't x1 be (x^2 + x^3 + .. + x^12) because it can only go as high as 12 right, not go on forever?
There's no difference in the end, since the coefficients will still be the same even if we allow it to go higher. (This doesn't generalize to all questions)
Awesome! Thanks for the video, it really helps me understand the topic
great explanation sir
Hi, Mr Trev, I have a question regarding the jellybean question.In the first conditon, it says even reds, but the second conditon says AT LEAST 14, does it give the restriction to the first condition? which in my assumption, its: 1 + x2 + x4 +x6. Corret me if I am wrong? thanks a lot
thats true, however, when you count the coefficient of x^20 none that had reds greater than x^6 included, so it doesnt affect the final solution
haha 2 years later
@@unusualgaming8629 ....a question being answered after 2 years...lol
@@eddiechen6389 and how can we count the coefficient of x^20 ?
@Abderrahim Mougari that’s 1
you saved my life.....love from india
Please can you create a playlist of your videos or numbering them so I can follow it as a course.
Thank you for a helpful videos
Playlist already exists for Discrete 1 and Discrete 2 series.
17:54 how to solve it further. I have watched complete series of generating functions mentioned on your channel. Is it the case of partial fraction decomposition and solve for each a0/(1-ax)^n ? Please reply m stuck.
what will be X^12 at 2:30?
i actually don't really understand but it help a litle bit. thank you
Thanks for the video, helped a lot! Stay Awesome!
This video is really helpful
Thanks a looooot for making this video!!!
Regarding long division of polynomials - aren't you suppose to line them up in decreasing order, i.e. -x + 1 instead of 1-x. How does this affect the answer? Also the 1/(1-x) solution is only true for |x| < 1. Are we assuming this?
you are simply awsome man !!!!
So, if I understand correctly, there are 12 ways to solve the first question where x1+x2+x3=12 with 0≤xi≤6. You said that there are 3+4+5 ways of solving the equation, so that would be 12 ways in total, right?
It really helps a lot!!!
Thanks!
Is the way you are using the exponent syntax different from the traditional definition of repeated multiplication?
I really don't like discrete maths :D
+Qlimakz That's okay. It's an infinitely smaller subset of continuous math, so you're not hurting its feelings too much.
That why they call it "Dry Subject".Fucks with your logical part of your brain.
Yeah man...i wish it just dissapears...
It's my favourite subject
@@kapilkumarsharma4401 Same
At 6:02 you didn't show how to get the coefficients of x^n
This is super goood 😀😀
4:17 how can we have more than x^6 for red if we have at least 14 blue jellybeans?? wouldnt that make it more than 20?
For the final problem, why isnt the x_1 term (1-x^11)/(1-x)? Isnt it the sum of values from 1 to x^12 by the problem statement?
Why for picking even ones we consider X^0 wich is 1 also as an even number? If we have an empthy set or something that is zero we consider it even?
Hello, what software do you use for writing with the graphical tablet?
Awesome...Thanks for the video...
thank you so much for your videos! Got a combinatorics puzzle. Could you please help me with it? Choose 6 numbers from 1 to 49 so that exactly two of them are consecutive. How many combinations?
I FOUND A MISTAKE !!!!
in the question of red,blue and white
shouldn't the red be (1+x^2+x^4+x^6) that's it
because we have to give atleast 14 to the blue one,then how the red can go upto ...+x^20
expecting a reply soon
thanks :)
It doesn't make a different. In the end we look at the coefficient of x^20. Letting red go up to x^20 won't affect the overall coefficient of x^20 after multiplying all the terms out.
yes sir
thanks
dude Ur awesome...thanks for the help
sir at 14:05 it will be 0,0,0,0,1,1,1,1....
THANK YOU!!!!!!!!!!!
thank you very much for your help
thank You I really Enjoy That man :))))
Me too
What if I wanted to generate polynomials? For example find the partition of a polynomial in the form of a pair of polynomials?
Thank great explanation
In the jellybeans example ,isn't the generating function (1+x^2+x^4+x^6)(x^14+x^15+x^16+x^17+x^18+x^19+x^20)(1+x+x^2+x^3+x^4),since we have at least 14 blue jellys ?
you don't need to stop at x^6 in this first term because any number above 6 will never be chosen since we will need to add it to at least 14 in the second term
Thank you soo much..
Hey sir, what software u used to write this?
why do we need the coefficient of x^12, Im not getting this part, explain plzzzz
Why did you change the x^2 to to x^5 in the last example?
multiplied x^2 and x^3 together.
I watching this vid two times ...then i realize ...its super easy 😂😂 ..my lecturer cant explain this like you do..😅
How can you use generating functions to find permutations?
2:25 you're right about my exes mister
Thank you so much!
What is the use of 12 in the second line
Thanks a lot
thank you
So helpful
These are nothing but G.P. . Sum of infinte G.P = a/1-r, but for that we've to assume that |x| < 1. Can we do that , is it the legit way to do ? As x is a variable of no significance , can we assume its value and solve the G.P. ?
what software do you use for your videos?
yogurt1989 Windows Journal to write, and either OBS or Camtasia to record the screen region.
It is interesting function. I think the x range should have range. 1/(1-x) forms you explained in the long division but when we divide 1 by 1-x the remnant x must be smaller than 1-x therefore, the x range should fall x
How is that (1-x) goes 1 time into 1, x times into x and x² times into x², I'm not getting the meaning of any of these yet you say it like it's super obvious
He's just matching the power of the (1) in (1-x) with the highest power of the polynomial. When the number is 1, you subtract a 1-x from 1 to get x. Then you subtract x(1-x) from x to get x^2. You continue this to reach the infinite series 1+x+x^2+x^3.... The principle is similar to long division, which is why it is called polynomial long division.
Peter i was taking calc 2 when i asked the question a few months ago. I get it now, thanks
I don't like it, I love it, love it, love it, uh oh
So good I found it!
How about exponential generating functions?
Is there a intro video to this that I missed? None of this makes sense....
It starts making sense after you watch the whole video.
3:49 almost 🇫🇷
shouldn't the first part of the last question be (x^2 * (1-x^13)/(1-x)) instead?
Thanks buddy
(1+x²+x⁴+....)⁷ we have got X^n coefficent please tell me sir
rolling a die 5 times how many ways we get sum of 18
You just turned confusing things , into even more confusing things
I regret my life because I had to watch this video 😥❤
Not all heroes wear capes😊