${\displaystyle {\begin{aligned}N&\in 0,1,2,\dots \\m&\in 0,1,2,\dots ,N\\n&\in 0,1,2,\dots ,N\end{aligned}}}$

${\displaystyle k\,\in \,\max {(0,\,n+m-N)},\,\dots ,\,\min {(m,\,n)}}$

${\displaystyle {\frac {{m \choose k}{N-m \choose n-k}}{N \choose n}}}$

${\displaystyle {\frac {nm}{N}}}$

${\displaystyle \left\lfloor {\frac {(n+1)(m+1)}{N+2}}\right\rfloor }$

${\displaystyle n(m/N)(1-m/N)(N-n)/(N-1)}$

${\displaystyle {\frac {(N-2m)(N-1)^{\frac {1}{2}}(N-2n)}{[nm(N-m)(N-n)]^{\frac {1}{2}}(N-2)}}}$

${\displaystyle \left[{\frac {N^{2}(N-1)}{n(N-2)(N-3)(N-n)}}\right]}$ ${\displaystyle \cdot \left[{\frac {N(N+1)-6N(N-n)}{m(N-m)}}\right.}$ ${\displaystyle +\left.{\frac {3n(N-n)(N+6)}{N^{2}}}-6\right]}$

${\displaystyle {\frac {{N-m \choose n}\ _{2}F_{1}(-n,-m;N-m-n+1;e^{t})}{N \choose n}}}$

${\displaystyle {\frac {{N-m \choose n}\ _{2}F_{1}(-n,-m;N-m-n+1;e^{it})}{N \choose n}}}$