What is the optimal number of dice to roll a Yahtzee in one roll?

By inclusion-exclusion, the full probability of Yahtzee is: $$\frac{1}{6^n}\sum_{k=1}^{\min(6,n/5)} (-1)^{k+1} \binom{6}{k} (6-k)^{n-5k} \prod_{j=0}^{k-1} \binom{n-5j}{5}.$$ If you prefer, write the product with a multinomial: $$\prod_{j=0}^{k-1} \binom{n-5j}{5}=\binom{n}{5k}\binom{5k}{5,\dots,5}.$$ Looks like $n=29$ is the uniquely optimal number of dice: \begin{matrix} n &p\\ \hline 28 &0.71591452705020 \\ 29 &0.71810623718825 \\ 30 &0.71770441391497 \\ \end{matrix}

Here is the SAS code I used:

proc optmodel;
   set NSET = 1..100;
   num p {n in NSET} = 
      (1/6^n) * sum {k in 1..min(6,n/5)} (-1)^(k+1) 
      * comb(6,k) * (if k = 6 and n = 5*k then 1 else (6-k)^(n-5*k)) 
      * prod {j in 0..k-1} comb(n-5*j,5);
   print p best20.;
   create data outdata from [n] p;
quit;

proc sgplot data=outdata;
   scatter x=n y=p;
   refline 29 / axis=x;
   xaxis values=(0 20 29 40 60 80 100);
run;

As an alternative approach, we can use the symbolic method to deduce that the generating function for the class of all rolls not containing a Yahtzee is given by

$$ f(z) = (e^z - z^5/5!)^6 $$

while the generating function for all rolls is

$$ g(z) = (e^z)^6. $$

The probability that a roll of $n$ dice yields a Yahtzee is given by

$$ 1-[z^n]f(z)/[z^n]g(z). $$

Using Mathematica:

f[z_] := (Exp[z] - z^5/5!)^6;
g[z_] := Exp[z]^6;
ans[n_] := 
  1 - SeriesCoefficient[f[z], {z, 0, n}]/
    SeriesCoefficient[g[z], {z, 0, n}];
DiscretePlot[ans[n], {n, 10, 40}]