If three moles of monoatomic gas $$\left(\gamma = \frac{5}{3}\right)$$ is mixed with two moles of a diatomic gas $$\left(\gamma = \frac{7}{5}\right)$$, the value of adiabatic exponent $$\gamma$$ for the mixture is:

For a mixture of gases, the effective $$\gamma$$ is found using:

$$\frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1} = \frac{n_1 + n_2}{\gamma_{mix} - 1}$$

$$\frac{3}{5/3 - 1} + \frac{2}{7/5 - 1} = \frac{5}{\gamma - 1}$$

$$\frac{3}{2/3} + \frac{2}{2/5} = \frac{5}{\gamma - 1}$$

$$\frac{9}{2} + 5 = \frac{5}{\gamma - 1}$$

$$\frac{19}{2} = \frac{5}{\gamma - 1}$$

$$\gamma - 1 = \frac{10}{19}$$

$$\gamma = 1 + \frac{10}{19} = \frac{29}{19} \approx 1.526$$

The answer is Option (3): $$\boxed{1.52}$$.