  We investigate the existence of normalized ground states to the system of
coupled Schr\"odinger equations: \begin{equation}\label{eq:0.1}
  \begin{cases}
  -\Delta u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{p_1-2}u_1 + \beta
r_1|u_1|^{r_1-2}u_1|u_2|^{r_2} & \text{ in } \mathbb{R}^{3},
  -\Delta u_2 + \lambda_2 u_2 = \mu_2|u_2|^{p_2-2}u_2 + \beta
r_2|u_1|^{r_1}|u_2|^{r_2-2}u_2 & \text{ in } \mathbb{R}^3,
  \end{cases}
  \end{equation} subject to the constraints $\mathcal{S}_{a_1} \times
\mathcal{S}_{a_2} = \{(u_1 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_1^2 dx
= a_1^2\} \times \{(u_2 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_2^2 dx =
a_2^2\}$, where $\mu_1, \mu_2 > 0$, $r_1, r_2 > 1$, and $\beta \geq 0$. Our
focus is on the coupled mass super-critical case, specifically, $$\frac{10}{3}
< p_1, p_2, r_1 + r_2 < 2^* = 6.$$ We demonstrate that there exists a
$\tilde{\beta} \geq 0$ such that equation (\ref{eq:0.1}) admits positive,
radially symmetric, normalized ground state solutions when $\beta >
\tilde{\beta}$. Furthermore, this result can be generalized to systems with an
arbitrary number of components, and the corresponding standing wave is
orbitally unstable.
