To solve this problem, we need to use the concept of half-life in radioactive decay.$\\$Given:$\\$• Half-life $(t_{1/2}) = 10$ minutes$\\$• Initial number of nuclei $(N_0) = 600 \\$• Number of nuclei disintegrated $= 450 \\$First, calculate the remaining number of nuclei after disintegration:$\\ N = N_0 - 450 = 600 - 450 = 150 \\$The formula for radioactive decay is:$\\ N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \\$Substitute the known values:$\\ 150 = 600 \left(\frac{1}{2}\right)^{\frac{t}{10}} \\$Divide both sides by 600:$\\ \frac{150}{600} = \left(\frac{1}{2}\right)^{\frac{t}{10}} \\ \frac{1}{4} = \left(\frac{1}{2}\right)^{\frac{t}{10}} \\$Recognize that $\frac{1}{4} = \left(\frac{1}{2}\right)^2 \\$Therefore,$\\ \left(\frac{1}{2}\right)^2 = \left(\frac{1}{2}\right)^{\frac{t}{10}} \\$Equate the exponents:$\\ 2 = \frac{t}{10} \\$Solve for $t: \\ t = 2 \times 10 = 20$ minutes$\\$Thus, the time taken for the disintegration of 450 nuclei is $20$ minutes.$\\$This corresponds to Option 3.