includeRow[Ri_, r_] := Mod[Floor[Ri/r], 2]; 
satisfyingRows[Ri_, s_] := 
      Switch[s, 0, includeRow[Ri, 1], 1, 
        includeRow[Ri, 2] + includeRow[Ri, 4] + includeRow[Ri, 16], 2, 
        includeRow[Ri, 8] + includeRow[Ri, 32] + includeRow[Ri, 64], 3, 
        includeRow[Ri, 128]]; 
c[n_, k_] := 
        Factorial[n]/(Factorial[k]*Factorial[n - k]);
sat[Ri_] := 
        Sum[(satisfyingRows[Ri, s]/c[3, s])*c[k, s]*c[n - k, 3 - s], {s, 
              0, 3}]/c[n, 3];

sat[22] /. n -> 1000 /. k -> 333

N[%]

sat[22]

sat[22] /. n -> 10 /. k -> 3

Expand[sat[22]]

c[n - k, 3 - s]

sat[Ri_] := 
        Sum[q[Ri, s]*((c[k, s]*c[n - k, 3 - s])/c[n, 3]), {s, 0, 
            3}];

sat[22] /. n -> 10 /. k -> 3

((3 k! (-3 + n)! (-k + n)! q[22, 1])/((-1 + k)! n! (-2 - k + n)!))

Simplify[%]

appSAT[22]

3 (((1 - x))^2) x /. x -> 3/10

satisfyingRows[22, 3]

((c[k, s]*c[n - k, 3 - s])/c[n, 3]) /. n -> 10 /. 
      k -> 3

c[k, s]

c[n - k, 3 - s]

c[n, 3]

ExpandAll[c[n, 3]]

c[k, s]*c[n - k, 3 - s]

FullSimplify[c[n - k, 3 - s]]

FullSimplify[c[n - k, 3 - s],  
      n > 0 && 0 ≤ k ≤ n && 
        0 ≤ s ≤ 3 && Element[n, Integers] && 
        Element[k, Integers] && Element[s, Integers]]

sat[22]

FullSimplify[sat[r],  
      n > 0 && 0 ≤ k ≤ n && Element[n, Integers] && 
        Element[k, Integers]]

% /. n -> 1000 /. k -> 300

% /. r -> 15

(3*k*c[n - k, 2])/c[n, 3]

FullSimplify[(3*k*c[n - k, 2])/c[n, 3],  
      n > 0 && 0 ≤ k ≤ n && Element[n, Integers] && 
        Element[k, Integers]]

(3*300*c[1000 - 300, 2])/c[1000, 3]

appSAT[22] /. x -> (k/n)

((9 k (k - n) (1 + k - n))/(n (2 - 3 n + (n^2)))) == ((3 k (((1 - (k/n)))^2))/n)

ForAll[{n, 
        k}, ((9 k (k - n) (1 + k - n))/(n (2 - 3 n + (n^2)))) == ((3 k (((1 - (k/n)))^2))/n)]

\(Reduce[∀ \_{n, k}\(9\ k\ \((\(-n\) + k)\)\ \((1 - n + k)\)\)\/\(n\ \((2 - 3\ n + n\^2)\)\) == \(3\ k\ \((1 - k\/n)\)\^2\)\/n, {n, k}, Integers]\)

3*(7^3/10^3)

3*(1 - x)^2*x /. x ->  .3

Expand[3*(1 - x)^2*x]

(3*k*c[n - k, 2])/c[n, 3] /. k -> n/3

Limit[%,n->Infinity]