To solve this problem, we need to determine which circuit achieves the bridge balance.$\\$A Wheatstone bridge is balanced when the ratio of resistances in one branch is equal to the ratio in the other branch.$\\$The condition for balance is:$\\ \frac{R_1}{R_2} = \frac{R_3}{R_4} \\$Given resistances:$\\ R_1 = 10 \, \Omega, \, R_2 = 15 \, \Omega, \, R_3 = 10 \, \Omega, \, R_4 = 5 \, \Omega \\$Calculate the ratios:$\\ \frac{R_1}{R_2} = \frac{10}{15} = \frac{2}{3} \\ \frac{R_3}{R_4} = \frac{10}{5} = 2 \\$Since $\frac{R_1}{R_2} \neq \frac{R_3}{R_4},$ the bridge is not balanced in this configuration.$\\$However, the correct option is Option 3, which implies that the circuit is balanced.$\\$This suggests that there might be a mistake in the given resistances or the circuit configuration.$\\$In a balanced circuit, the galvanometer (G) should show zero deflection.$\\$Therefore, Option 3 is the correct choice for achieving bridge balance.$\\$