The correct answer is c) 1, 3 and 4 only.
The Muhatsibs were officials in Muslim societies who were responsible for enforcing religious and moral laws. Their duties included checking and regulating various aspects of public life, including gambling dens, weights and measures, and the consumption of intoxicants in public places.
Option a) 1 and 2 only is incorrect, because the Muhatsibs did not have the duty to check and interfere in the private life of the citizens. Their role was to enforce religious and moral laws in public spaces, not to intrude on the private lives of individuals.
Option b) 1, 2, 3 and 4 is incorrect, because the Muhatsibs did not have the duty to check and interfere in the private life of the citizens.
Option d) 3 and 4 only is incorrect, because the Muhatsibs did have the duty to check gambling dens, in addition to checking weights and measures and enforcing laws against the consumption of intoxicants in public places.
Overall, the correct answer is c) 1, 3 and 4 only, because these are the duties that were actually assigned to the Muhatsibs.