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Mirzakhani's asymptotic formula for the number of simple closed geodesics of length $\\le L$ on a hyperbolic surface of genus $g$ with $n$ punctures. We investigate the number of simple closed geodesics of length $\\le L$ representing a fixed primitive nonzero homology class $x$ on a hyperbolic surface $S$. We denote this number by $h_{S}(L, x)$. It follows from Mirzakhani's result that $h_{S}(L, x) \\le C L^{6(g-1) + 2n}$. 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