## Prove that the likelihood function is a sufficient statistic

Let $f(x|\theta)$ be a family of densities where the parameter space is finite, i.e., $\theta \in \Theta = \{\theta_1, \cdots, \theta_p\}$. Now consider the likelihood function statistic, defined to be $T(\mathbf{X}) = (f_{\theta}(\mathbf{X}))_{\theta \in \Theta} = (T_1(\mathbf{X}), \cdots, T_p(\mathbf{X})) = (f_{\theta_1}(\mathbf{X}), \cdots, f_{\theta_p}(\mathbf{X}))$. Show that $T(\mathbf{X})$ is a sufficient statistic for $\theta$ but not a minimal sufficient statistic.

Do I use factorization theorem to show it is sufficient? I am unsure of how to start. Any help would be appreciated!