[T0rajir0u]

lmao i got it after 2 mins in the shower

and what are you talking about the combinatorial argument is hella easy but i got trixxed thinking about the squares

anyway

(nCk)^2 = (nCk)(nC(n-k))

RHS is the number of ways to pick n objects out of 2n objects

you can do this by picking k objects out of the first n objects and then n-k objects out of the second n for k = 0... n and that gets you the LHS

induction is retarded 4 dis problerm LETS NOT EVEN GO THERE

and geometry wtf i will keep thinking about that

EDIT: lmao 5 more minutes trying to fall asleep and i have another counting argument

this time its path counting

basically we're tracing paths down pascal's triangle (easy way to think about it) and every entry in the triangle tells you how many paths there are from the top to that entry

we want to find the number of paths to the middle entry in the 2nth row

so we trace out all the possible paths to the nth row and then there's a nice symmetry between the first n rows of the journey and the second n rows that lets us conclude that the number of paths to the 2nth row passing through the kth entry of the nth row is just (nCk)^2 (there are nCk paths to the kth entry of the nth row and because the 2nth row and 0th row are symmetric about the nth row there are nCk paths from the kth entry of the nth row to the nth entry of the 2nth row)

sum it up etc

pretty similar to the other argument but i don't think pathcounting = geometry so lemme think about it some more