Consider Grassmannian of $d$-dimensional subspaces of $n$-dimensional vector space
$V$. Then you have Pluker inbeding of grassmanian $G=Gr(d,V)$ of d-planes of $V$ in $\mathbb P(\Lambda ^d V)=\mathbb P^N, N=\binom{n}{d}-1$, as follows. For every $d-plane$ D chose base $v_1,...,v_d$ of $D$ and consider exterior product $\omega = v_1\wedge...\wedge v_d \in \wedge ^d V$. Pluker inbeding send $D$ to line: Span $\omega \in \mathbb P(\Lambda ^d V)$ .
For find Zariski covering, fix base $e_1,...,e_n$ of $V$ and receive base $e_I=e_{i_1}\wedge...\wedge e_{i_d}$ of $\wedge ^d V $ ($I$ is increasing sequence $I=(1\leq i_1 \lt ...\lt i_d \leq n$)). Zariski covering consist of $U_I$ formed by these $d-$ planes $D$ whose Pluker image in $\mathbb P(\wedge ^d V)$ has coordinate $z_I\neq 0$. This mean that when you chose base $v_1,...,v_d$ of $D$ and developp $\omega = v_1\wedge...\wedge v_d \in \wedge ^d V$ in base $e_I$ of $\Lambda ^d V$, then coefficient of $e_I$ is non-null. This Zariski cover $U_I$ of grasmanian $Gr(d,V)$ possesses $N+1=\binom{n}{d}$ open sets (Attention: $\mathbb P^N$
has $N+1$ coordinates, not $N$)