Position of Indices in Einstein's Summation Convention Would there be a difference between the following tensor quantities?
$$ {A_\gamma}^\mu {B_\mu}^\rho $$ and $$ {A^\mu}_\gamma {B_\mu}^\rho $$
Would both of these expressions give the same result as follows
$$ {X_\gamma}^\rho = {A_\gamma}^\mu {B_\mu}^\rho = {A^\mu}_\gamma {B_\mu}^\rho? $$
As the positions of the indices in Einstein's summation convention make a difference, I believe that the two expressions are not equivalent. The first equation $ {X_\gamma}^\rho = {A_\gamma}^\mu {B_\mu}^\rho $ makes sense to me as the positions of the indices $\gamma$ and $\rho$ are clear in the resultant quantity ${X_\gamma}^\rho$ but I am unsure of the second equation. Would $ {X_\gamma}^\rho = {A^\mu}_\gamma {B_\mu}^\rho $?
 A: 
Would there be a difference between the following tensor quantities?

Yes. Your equation is false. The l.h.s. contracts the second index, and the r.h.s. one contracts the first one (even if you are using the same letter for the index). Unless your tensors have some sort of symmetry, these expressions do not agree.
A: Let's start with the metric $\eta_{\mu\nu}$ such that $\Delta s^2 = \Delta x^\mu \; \eta_{\mu\nu} \; \Delta x^\nu$. Denote the inverse metric as $\eta^{\mu\nu}$ so that $\eta_{\mu\nu}\eta^{\nu \rho} = \delta_{\mu}^{\,\,\rho}$.
Start with tensor with only index up, like
$$
A^{\gamma\mu} \qquad \qquad \text{and} \qquad \qquad B^{\mu \rho}
$$
then define the lower-index version of these tensors by contracting with the metric $\eta^{\mu\nu}$:
$$
A_{\gamma}^{\;\,\mu} := \eta_{\gamma \alpha} A^{\alpha \mu}; \qquad \qquad
A_{\,\;\mu}^{\gamma} := \eta_{\mu \alpha} A^{ \gamma\alpha}
$$
and so on.
You see that for instance
$$
X_{\gamma}^{\;\;\rho} = A_{\gamma}^{\;\;\mu} B^{\;\;\rho}_{\mu} \equiv 
\eta_{\gamma \alpha} \eta_{\mu\beta} A^{\alpha\mu}B^{\beta\rho}
$$
whereas
$$
Y_{\gamma}^{\;\rho} = A_{\;\;\gamma}^{\mu} B^{\;\;\rho}_{\mu} \equiv 
\eta_{\gamma \alpha} \eta_{\mu\beta} A^{\mu\alpha}B^{\beta\rho}
$$
and you see that are, in general, two different tensors, and only if $A$ (in this case) is symmetric they are equal. 
But note that calling the first quantity $X_{\gamma}^{\;\;\rho} $ or$ X_{\;\;\gamma}^{\rho} $ is only a matter of definition, they would both be consistent with index conventions.
