Solutions to 3b1b's generating function exercises: from rolling dice to differential equations

In his recent video on an "innocent" enumeration problem, @3blue1brown offered four problems on generating functions. In this video, we discuss these problems, add a bit of context here and there, and go over detailed solution suggestions. Enjoy!

#math #combinatorics #generatingfunctions #visuallyexplained #discretemathematics #manim #solution #solutions #counting #dice #fibonacci

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### Links to resources

### More about me

### Attributions

### Music

### Contents

0:00 Introduction
1:30 The Generating Functions and the Die
8:25 A Binomial Sum
11:20 Fibonacci and a Differential Equation
20:52 A useful Product
24:04 Outro

[andyblackett]

This is great, I am an A Level Maths teacher in UK, I have a student who will have watched the 3B1B video and will come to me on Monday asking about the 4 questions at the end - so it's something of a lifesaver for me! Many thanks.

[apotato]

3B1B's video was fascinating on its own, and even gave the inspiration that finally solved one of my unsolved problems from a couple months ago. However, this video clearly solidifies interesting uses that generating functions have. I found the part about the fibonacci numbers especially intriguing, probably since I've known about the numbers themselves and seen the closed form solution before. And then suddenly your derivation of it only uses a bunch of math I already know! And, sure, I might not have figured out that you could combine each of the steps in this way, but each individual step is something that I could have solved given some guidance. I will remember that one in the future, for sure.

[MathVisualProofs]

Excellent stuff here! I have been wanting to do something like this but I couldn’t have done it like this. Very cool :)

[Cooososoo]

Yes I am curious about generating function and Thnx 🎉

[PowerhouseCell]

Woah, this is such an underrated channel! As a fellow educational YouTuber, I understand how much work must have gone into this— amazing job!! Liked and subscribed :)

[SamirSelim]

what are programs did you use in this video, sir? nice presentation

[asrafialmamun]

How do you show your face on the screen while recording videos?

[rons.]

In problem 1 the expected number of dice rolls is 6. Your expected value refers to the number of "previous rolls" so you gotta add 1. You didn't say why you focused on previous rolls. Great explanations otherwise.