Exercise. Prove that the binomial coefficient (n,k) is an integer. Equivalently, show that the product of k consecutive integers is divisible by k! . Prove this equivalence.
6:18 i was confused for a bit about those eights but then remembered it's possible to exclude numbers starting with 0 because you're looking for n-digit numbers, not strings of n digits
At 10:00 you start turning each of the S_i subsets (for I in {0..n}) into lists. I found this confusing for two reasons: 1) I did not see a motivation for this idea, 2) you use S_i to refer to the new lists-but S_i is an (unordered) set, you cannot “reorder” it. I guess that when you add a superscript, you are converting S_ i into a list, S_i^j where j in {1..k!}. BTW I am writing “1..n” for “1,...,n”.
Didn’t think I’d ever like the math your videos cover as an applied physicist by trade. I’m referring more to your analysis and number theory videos. There needs to be more profs like you.
Hi, For fun: 4 "let's may be go ahead and", 1 "let's go ahead and", 1 "great", 1 "now next, what I want to do", 1 "and so on and so forth", 3:36 : error in the position of the first comma.
Hi Michael, as always great and instructive job! Regarding combinatorics, it would be worth giving a proof of IMO 2014/6 ... I couldn't find only a very clear proof indeed. Thanks and stay safe out there!
n and k should be nonnegative integers. The definition of the binomial for negative integers isn't well defined from what you wrote in the theorem. And isn't the factorial definition better if k > n? In n(n-1)..(n-k+1)/k! you'd have 1/k! if you interpret the numerator as an empty product. n!/(k!(n-k)!) is at least undefined (and using the gamma function, a factorial of a negative integer is infinity giving 0 as desired).
This doesn’t take away from the theory, but not all letters are used in license plates. For example O is never used to avoid confusion with the number 0
Is that in Virginia? What about I (the 9th letter) and the number one? Here in the UK (where we have a national system rather than individual state systems) we have a bunch of weird rules - Z are kept for Northern Ireland only, Q is used for cars imported from elsewhere, letters I and O are similarly not used, but we allowed U when it was in the main letter region, but not in the year identifier (so you could have UUK 144T (T was March 1999) but the same plate the following year was UUK 144V (V was Sept 1999))... it got weirder when they changed things again in 2001. In the Isle of Man, all number plates were MAN - a total of one possible combination of letters.
Love the minimalistic titling.
I should mention that I really enjoy your videos. No fluff, straight to the working out.
Please, more combinatorics!
Your videos are helping a lot for preparing for math competitions. Keep doing it.
Your a legend.
17,5760,000
You are a great teacher Michael sir!👍👍❤️❤️❤️❤️
The examples are very interesting.
Exercise. Prove that the binomial coefficient (n,k) is an integer. Equivalently, show that the product of k consecutive integers is divisible by k! . Prove this equivalence.
6:18 i was confused for a bit about those eights but then remembered it's possible to exclude numbers starting with 0 because you're looking for n-digit numbers, not strings of n digits
at 5:53 he actually says this out loud, though it's easy to miss
where can I find the entire course of intro to proof writing?
At 10:00 you start turning each of the S_i subsets (for I in {0..n}) into lists. I found this confusing for two reasons: 1) I did not see a motivation for this idea, 2) you use S_i to refer to the new lists-but S_i is an (unordered) set, you cannot “reorder” it. I guess that when you add a superscript, you are converting S_ i into a list, S_i^j where j in {1..k!}. BTW I am writing “1..n” for “1,...,n”.
Didn’t think I’d ever like the math your videos cover as an applied physicist by trade. I’m referring more to your analysis and number theory videos. There needs to be more profs like you.
18:16 Oh, so you like counting... name every number then
O I see you love Trigonometry and Algebra and Geometry. List every write every digit of π then.
@@The_Math_Enthusiast You like geometry? Name all the polygons!
@@goodplacetostop2973 I see you love math, list every digit of e then.
Can you list every good place to stop?
@@ahzong3544 All I know is that is countable, like the rooms of Hilbert's hotel
At 6:22 (*)(*)(3) should be 8*8*9 because 003 is not allowed. It would be a single digit number not a three digit number.
Hi,
For fun:
4 "let's may be go ahead and",
1 "let's go ahead and",
1 "great",
1 "now next, what I want to do",
1 "and so on and so forth",
3:36 : error in the position of the first comma.
Be safe and enjoy your life
🙏
more combinatorics please!
very useful, thank you
Hi Michael, as always great and instructive job! Regarding combinatorics, it would be worth giving a proof of IMO 2014/6 ... I couldn't find only a very clear proof indeed. Thanks and stay safe out there!
3:15 my head twitches when he says 27 and writes 26 on the board.
And then 4:55 saying 143 pointing *3*...
There is a nice counting argument for why (n,0)+(n,1)+....+(n,n)= 2^n, here (a,b) means a choose b
At 10:51 you say “possible sets” when you mean “possible lists”.
an interesting one
Based on the title, this is probably too advanced for me
n and k should be nonnegative integers. The definition of the binomial for negative integers isn't well defined from what you wrote in the theorem.
And isn't the factorial definition better if k > n? In n(n-1)..(n-k+1)/k! you'd have 1/k! if you interpret the numerator as an empty product. n!/(k!(n-k)!) is at least undefined (and using the gamma function, a factorial of a negative integer is infinity giving 0 as desired).
This doesn’t take away from the theory, but not all letters are used in license plates. For example O is never used to avoid confusion with the number 0
Is that in Virginia? What about I (the 9th letter) and the number one? Here in the UK (where we have a national system rather than individual state systems) we have a bunch of weird rules - Z are kept for Northern Ireland only, Q is used for cars imported from elsewhere, letters I and O are similarly not used, but we allowed U when it was in the main letter region, but not in the year identifier (so you could have UUK 144T (T was March 1999) but the same plate the following year was UUK 144V (V was Sept 1999))... it got weirder when they changed things again in 2001. In the Isle of Man, all number plates were MAN - a total of one possible combination of letters.
At 9:15 (plus / minus some seconds) it should be written (n-k)! not (n-k!) (In denominator)...
i think it should be (n-k)! instead of (n-k!)
i watch all
27!? The comma!? Lol.
At 6:58 it should have “(n-k)!” instead of “(n-k!)”. For the second bulleted item.
While humans are presenting there will always be a few mistakes etc. The key is to spot them and so confirm understanding.