For the redox reaction :

${\mathrm{MnO}}_4^{-}+{\mathrm{Fe}}^{2+}+{\mathrm{H}}^{+}⟶{\mathrm{Mn}}^{2+}+{\mathrm{Fe}}^{3+}+{\mathrm{H}}_2\mathrm{O}$

the correct stoichiometric co-efficients of ${\mathrm{MnO}}_4^{- }, {\mathrm{Fe}}^{2+}$ and ${\mathrm{H}}^{+}$ are :

$\underline{\begin{array}{r}{\mathrm{MnO}}_4^{-}+8{\mathrm{H}}^{+}+5{\mathrm{e}}^{-}⟶{\mathrm{Mn}}^{2+}+4{\mathrm{H}}_2\mathrm{O}\\ {\mathrm{Fe}}^{2+}\left.⟶{\mathrm{Fe}}^{3+}+{\mathrm{e}}^{-}\right]\times 5\end{array}}$

Adding ${\mathrm{MnO}}_4^{-}+5{\mathrm{Fe}}^{2+}+8{\mathrm{H}}^{+}⟶{\mathrm{Mn}}^{2+}+5{\mathrm{Fe}}^{3+}+4{\mathrm{H}}_2\mathrm{O}$

                                                                                        …(1)

$\therefore$ Correct stoichiometric co-efficients of ${\mathrm{MnO}}_{4,}^{-}$ ${\mathrm{Fe}}^{2+}$ and ${\mathrm{H}}^{+}$ in the reaction (1) are, $1, 5, 8.$