Consider the following radioactive decay process:
$$^{218}_{84}A \xrightarrow{\alpha} A_1 \xrightarrow{\beta^-} A_2 \xrightarrow{\gamma} A_3 \xrightarrow{\alpha} A_4 \xrightarrow{\beta^+} A_5 \xrightarrow{\gamma} A_6$$
The mass number and the atomic number of $$A_6$$ are given by:

We need to track the mass number (A) and atomic number (Z) through a series of radioactive decays starting from $$^{218}_{84}A$$. Recall the effects of each type of decay: - Alpha decay ($$\alpha$$): Emits $$^4_2He$$. Mass number decreases by 4, atomic number decreases by 2. - Beta-minus decay ($$\beta^-$$): A neutron converts to a proton. Mass number unchanged, atomic number increases by 1. - Beta-plus decay ($$\beta^+$$): A proton converts to a neutron. Mass number unchanged, atomic number decreases by 1. - Gamma decay ($$\gamma$$): Only energy is emitted. Neither mass number nor atomic number changes.

Starting with $$^{218}_{84}A$$ (A = 218, Z = 84), after $$\alpha$$ decay ($$A_1$$) we get A = 218 - 4 = 214, Z = 84 - 2 = 82; after $$\beta^-$$ decay ($$A_2$$) A = 214, Z = 82 + 1 = 83; after $$\gamma$$ decay ($$A_3$$) A = 214, Z = 83; after $$\alpha$$ decay ($$A_4$$) A = 214 - 4 = 210, Z = 83 - 2 = 81; after $$\beta^+$$ decay ($$A_5$$) A = 210, Z = 81 - 1 = 80; finally, after $$\gamma$$ decay ($$A_6$$) A = 210, Z = 80.

This gives $$A_6$$ with mass number 210 and atomic number 80, so the correct answer is Option (3): 210 and 80.