Consider $N$ an integer and $k \in \{ 0, .., N\}$ and define $$ \Gamma_k := \{ \omega \in \{ 0,1 \}^N, \: \sum_{j=1}^N \omega_j = k \}. $$
The uniform law on $\Gamma_k$ satisfies $$ \forall \omega \in \Gamma_k,\: p(\omega) = \frac{k!(N-k)!}{N!} \: \mbox{(weight of $\omega$)}. $$
How do you simulate the uniform law on $\Gamma_k$?