# How to intelligently use FullSimplify and FunctionExpand to simplify complex sums

I am trying to find a compact form of some sums which is related with some Bayesian probability factor (not so relevant, if required further explanation please ask). The point is that I know that the following beast

Sum[((N1 - z1)! z1! (N2 - z2)! z2! (k2 + k3 + z01)! (k1 + k4 +
z10)! (-k2 - k4 + N1 + N2 + Nt - z01 - z1 - z10 - z11 -
z2)! (-k1 - k3 + z1 + z11 + z2)!)/(k1! k2! k3! k4! (N1 - z1 -
k2)! (z1 - k1)! (N2 - z2 - k4)! (z2 - k3)! (3 + N1 + N2 +
Nt)!), {k1, 0, z1}, {k2, 0, N1 - z1}, {k3, 0, z2}, {k4, 0,
N2 - z2}]


has a nice, compact form that is somehow SIMILAR (but not exactly equal) to

(z1! (N1 - z1)! z2! (N2 - z2)! z10! z01! z11! (Nt - z01 - z10 - z11)!)/(3 + N1 + N2 + Nt)!


Here, the following relations are satisfied. All variables are integers and:

$$0 \leq z1 \leq N1$$

$$0 \leq z2 \leq N2$$

$$0 \leq z11$$

$$0 \leq z01$$

$$0 \leq z10$$

$$z11 + z01 + z10 \leq Nt$$.

I have tried pretty much everything... simplifying all at once, take term by term, any combinations of them, Simplify, FullSimplify, FunctionExpand, with Assumptions, without assumptions...

I always manage to ALMOST arrive at the desired result, until Mathematica spits out a Hypergeometric function such as

HypergeometricPFQ[{-z1, 1 + z10, -1 - k2 - z01 - z1 - z11,
2 - k2 + N1 + N2 + Nt - z01 - z1 - z11 - z2}, {-z1 - z11,
2 - k2 + N1 + Nt - z01 - z1 - z11, -1 - k2 - z01 - z1 - z11 - z2},
1]


which Mathematica believes goes to ComplexInfinity under the aforementioned assumptions... Which is not mathematically correct.

At this point I do not know where the mistake is. As usual, I think that the problem is the person staring the computer, but I am starting getting quite desperate. If you have any suggestion, would be really great, Thanks!