**Question**: let $W\_1, W\_2, W\_3$ be three distinct subspaces of vector space $\mathbb{R^{12}}$ each having dimension $4$, and let $W=W\_1 ∩W\_2 ∩ W\_3$ then which of the following **is/ are** correct?

(1) $W$ is subspace of $\mathbb{R^{12}}$ 

(2) $dim(W)< 8$

(3) $dim(W)>7$

(4) $dim(w)<10$

**My attempt:** clearly $W$ will be subspace of $V$ (as intersection of subspaces is again subspace) hence **(1) is true**.

I stuck on other options, but I know $dim(W)≤dimV=12$. Further, I know if $W\_1, W\_2$ are subspaces of finite dimensional vector space then,

$dim(W\_1+W\_2)= dim(W\_1)+dim(W\_2)-dim(W\_1∩W\_2)$ 

But, **how to use** this formula, as we had intersection of three subspaces? further, using above formula is useful here? Please help needed.