This is coming up in a solution of a detailed balance equation. I'm fine with the theory, I'm just missing something silly here:

I have the detailed balance eqn:

πjuj = λ(K-(j-1))πj-1

So this turns into something like:

πj = (
λ/u) * (K-j+1) / j * πj-1

And then:
πj = \binom {K}{j} (\frac {\lambda}{u})^j π​0

Everything but the getting the iteration of
(K-j+1) / j to come out as the binomial is fine. I looked through all the various binomial formulas and tried to shoehorn that into the familiar K!/(j!(K-j))! assuming there's some equivalence in form I'm just not seeing (well, that's still true, as there must be).

Could someone please just point out what the silly thing I'm not seeing is?