Defining the inverse of a tensor via the adjugate tensor

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.