This is unnecessarily complicate: its far easier to reason as follows. If d_k is the number of derangements in k persons then n!= sum(C^n_k d_k, k=0 to n) as every permutation have exactly 0, 1,...,or n fixed points where C^n_k is the binomial number n over k. Now, its easy to show using formal series that for any sequence {a_k}_k it follows that b_n=sum( C^n_k a_k, k=0 to n) if and only if a_n=sum(C^n_k b_k (-1)^k, k=0 to n), so it follows in our case that d_n=sum(C^n_k k! (-1)^k,k=0 to n), and using the series of the exponential function this last result can be simplified to d_n=floor(1/2+n!/e). Tadaaaaa....! 😁
Deranged? More like mathematically misunderstood! Numbers like this are a reminder that math has a sense of humor.This video makes me want to brush up on my skills. SolutionInn has been my go to for tackling these unexpected surprises.
Deja vu or my other self in a parallel universe have already watched this video? I wish I can ask him but we still can't communicate.
one month ago, with a different example
As usual, another super wonderful math video!!! Also I Love your shirt!!! 🥰
Thank you!!?!
This is unnecessarily complicate: its far easier to reason as follows. If d_k is the number of derangements in k persons then n!= sum(C^n_k d_k, k=0 to n) as every permutation have exactly 0, 1,...,or n fixed points where C^n_k is the binomial number n over k. Now, its easy to show using formal series that for any sequence {a_k}_k it follows that b_n=sum( C^n_k a_k, k=0 to n) if and only if a_n=sum(C^n_k b_k (-1)^k, k=0 to n), so it follows in our case that d_n=sum(C^n_k k! (-1)^k,k=0 to n), and using the series of the exponential function this last result can be simplified to d_n=floor(1/2+n!/e). Tadaaaaa....! 😁
1854 is possibly the birth year of Sherlock Holmes.
What about the integral formula or using the nearest integer of n!/e?
!N ~ N!/e would have been a nice observation for closing 😊
I always forget if it is floor or round of that expression ;)
Deranged? More like mathematically misunderstood! Numbers like this are a reminder that math has a sense of humor.This video makes me want to brush up on my skills. SolutionInn has been my go to for tackling these unexpected surprises.
See the envelope stuffing problem 🙂