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A horizontal circular platform of radius 0.5 m and mass 0.45 kg is free to rotate about its axis. Two massless spring toy-guns, each carrying a steel ball of mass 0.05 kg are attached to the platform at a distance 0.25 m from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have horizontal speed of 9 ms–1 with respect to the ground. The rotational speed of the platform in rad s–1 after the balls leave the platform is:

Could you please tell how the net external torque for the system is zero because in one of its solutions they have applied conservation of angular momentum.

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If your system consists of the platform, the guns, and the balls, then the only external forces on this system are between the platform and whatever base it's mounted on. We are told that the platform is "free to rotate on its axis", which means that the torque exerted on the platform by the base is negligible. Thus, we can assume that the total external torque exerted on the system is negligible.

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