When Ron Goodwin set out to find an inexpensive place to stay while going to college here, he hardly expected to wind up living in a 35-room Chevy Chase manion. And he doesn't pay a dime.

"It's been pretty good," says Goodwin, 27, of Ft. Lauderdale, Fla., who is beginning his fourth year of a five-year program in pharmacy school at Howard University. Goodwin is one of the thousands of Washington-area college students who scramble each year to find off-campus housing in the city's tight rental market.

Not all of them, of course, live quite so luxuriously. Goodwin has his own room in the east wing of a Tudor landmark on Chevy Chase Circle and shares a bathroom with the butler and maid. He takes his meals with the family -- "nothing I could ever afford or have time to prepare," he says.

All of this he gets in exchange for working a maximum of 20 hours a week looking after his landlords 5-and-7-year-old daughters from after school until dinner. This includes such decidedly unarduous tasks as swimming with them at their country club pool. With this arrangement, he cuts the cost of his studies by several hundred dollars a semester.

Most of the area's large college campuses do not have enough residence halls to house all of their out-of-town students. But students can get help finding off-campus living quarters through college off-campus housing offices that provide a listing of rooms or apartments available or the names of other students seeking roommates. Often families with an unused room who could use some extra money to pay the utility bills phone the college to list it.

Students pore over the listings daily. At this time of year, say campus housing officials, listings are snapped up within a day or two. As many as 150 students a day check into the University of Maryland off-campus housing office, says the office's Marlo Zeroth. There is no charge to the student or the landlord for the service.

(University off-campus housing offices are: American University, 686-2080; Howard, 636-6131; Georgetown, 625-3026; George Washington, 676-7221; and Maryland, 454-3645.)

This year students will have trouble finding a room for less than $100 a month, say housing officials. Many students share houses to split the cost, but even that isn't cheap. A three-bedroom house for three students may begin at $600 a month, plus utilities, says Georgetown University's Maryjo Viscuso.

An alternative to the college housing offices is a new Washington firm, the National Student Housing Referral Service, Inc. (659-0755), established this year to take advantage of the demand for off-campus housing. It is aimed at primarily students seeking room and board in exchange for child-care or light housekeeping. Goodwin found his mansion through the referral service.

The family pays a one-time fee of $90 and the student a fee of $50 for a successful match, and both are questioned beforehand on personal preferences such as smoking and pets. One student made what seemed a perfect match with a family, until the cat walked in, says the firm's founder, Hank Wilson. "She was allergic to them."

A similar new enterprise is the Student Housing Exchange Service $323-8481), with representatives in the District and the Maryland and Virginia suburbs.

American students are often reluctant to live in family situations, says Wilson -- "A lot of them think you're putting them back into a parent situation" -- but foreign students accept it readily. Most of the demand for a live-in student is for females who will care for children. "It's an unfortunate stereotype that men can't take care of kids," says Ann Liebschutz, the service's director.

Wilson says Goodwin's isn't the only dream situation his firm has arranged. One student, he says, gets room and board just for driving the landlord's daughter to her ice-skating class and another for taking care of a private tennis court.