= Solution
The <torsion tensor> of an <affine connection> is $\mathcal T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]$, where the last term is the <Lie bracket of vector fields>. A <torsion-free connection> has $\mathcal T=0$. Since a <coordinate basis> commutes, this is equivalent there to $\Gamma^\mu{}_{\nu\rho}=\Gamma^\mu{}_{\rho\nu}$; in a general <basis> its commutator must also be included.
The <Levi-Civita connection> is the unique <torsion-free connection> with <metric compatibility>, $\nabla g=0$. To derive its <Christoffel symbols>, lower the first index: $\Gamma_{\sigma\nu\rho}=g_{\sigma\mu}\Gamma^\mu{}_{\nu\rho}$. <Metric compatibility> gives
$$
\partial_\rho g_{\sigma\nu}=\Gamma_{\nu\sigma\rho}+\Gamma_{\sigma\nu\rho}.
$$
Write down this identity with the derivative indices $\rho,\nu,\sigma$, add the first two identities and subtract the third. Symmetry in the last two indices, supplied by the <torsion-free connection>, cancels all terms except twice $\Gamma_{\sigma\nu\rho}$. Raising the first index yields
$$
\boxed{\Gamma^\mu{}_{\nu\rho}=\frac12g^{\mu\sigma}\bigl(\partial_\rho g_{\sigma\nu}+\partial_\nu g_{\sigma\rho}-\partial_\sigma g_{\nu\rho}\bigr).}
$$
This calculation proves uniqueness. Conversely, these <Christoffel symbols> are symmetric in $\nu,\rho$ and satisfy $\nabla g=0$ by substitution, proving existence as well.
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