Defining the inverse of a tensor via the adjugate tensor

Question

My professor definied the adjugate of a tensor $\mathbf{t}\in T^{1}_{1}(E)$ (E is just a vector space of dimension n) by defining its components as
$adj(\mathbf{t})^{a}_{b}=\frac{1}{(n-1)!}\varepsilon_{bi_{2}...i_{n}}t^{i_{2}}_{j_{2}}...t^{i_{n}}_{j_{n}}\varepsilon^{aj_{2}...j_{n}}$
Then he claims that $\mathbf{t} \circ adj(\mathbf{t})=det(\mathbf{t})\mathbf{id}_{E}$ and from this he defines the inverse tensor.
The point that I can't seem to prove is how we can obtain the expression of the determinant from the composition of the tensor with its adjugate. The expression for $ det(\mathbf{t})$ is basically just $\frac{1}{n!}\varepsilon_{i_{1}...i_{n}}t^{i_{1}}_{j_{1}}...t^{i_{n}}_{j_{n}}\varepsilon^{j_{1}...j_{n}} $ but I cannot seem to make this expression "appear" by multiplying the two matrices of the two endomorphisms associated with the tensors, it seems like I'm off by a factor of n somewhere, which should give me the $\frac{1}{n!}$ in front. Any help is appreciated.