The Finishing Trades Institute of New York will begin taking applications on Tuesday for 50 painter, decorator and paperhanger apprenticeships, officials said. Applications can be obtained at the institute's office, 45-15 36th St., Long Island City, Queens, from 9 a.m. to 3:30 p.m. on weekdays. Forms will be available until Aug. 29 or until 500 forms have been distributed, whichever comes first. Applicants must be 18 or older, have a high school diploma or its equivalent, be physically able to do the work, pass a drug test and be able to understand instructions in English. More information is available by calling 718-937-7440.