First, calculate the number of moles of methane ($\text{CH}_4$). Molar mass of $\text{CH}_4 = 12 + (4 \times 1) = 16\text{ g/mol}$. Moles of $\text{CH}_4 = \frac{\text{Given Mass}}{\text{Molar mass}} = \frac{1.6\text{ g}}{16\text{ g/mol}} = 0.1\text{ mol}$.

Next, calculate the number of molecules of methane in $0.1\text{ mol}$. Number of molecules = $\text{Moles} \times N_A = 0.1 \times (6.02 \times 10^{23}) = 6.02 \times 10^{22}$ molecules.

Now, determine the number of electrons in one molecule of $\text{CH}_4$. A carbon atom (atomic number 6) has $6$ electrons, and each hydrogen atom (atomic number 1) has $1$ electron. So, one $\text{CH}_4$ molecule contains $6 + 4(1) = 10$ electrons.

Finally, calculate the total number of electrons in $1.6\text{ g}$ of methane. Total electrons = Number of molecules $\times$ Electrons per molecule = $(6.02 \times 10^{22}) \times 10 = 6.02 \times 10^{23}$ electrons.