To determine the number of protons, neutrons, and electrons in $^{175}_{71}$Lu$,$ we need to understand the notation of isotopes.$\\$In the isotope notation $^{A}_{Z}$X$,$ $A$ is the mass number, $Z$ is the atomic number, and $X$ is the element symbol.$\\$For $^{175}_{71}$Lu$: \\$• $A = 175$ (mass number)$\\$• $Z = 71$ (atomic number)$\\$• Lu is the symbol for the element Lutetium.$\\$The atomic number $Z$ represents the number of protons. Therefore, $\\$• Number of protons $= 71 \\$In a neutral atom, the number of electrons is equal to the number of protons. Therefore,$\\$• Number of electrons $= 71 \\$The mass number $A$ is the sum of protons and neutrons. Therefore,$\\$• Number of neutrons $= A - Z = 175 - 71 = 104 \\$Thus, the number of protons, neutrons, and electrons in $^{175}_{71}$Lu$$ are 71, 104, and 71, respectively.$\\$This corresponds to Option 4.