Fleming International Airways, an all-cargo airline, walked away from a Federal Aviation Administration drawing yesterday with its first choice of Braniff International's valuable landing rights at seven airports, including Washington National and Dulles International.

Robert P. Fleming, the carrier's president, said yesterday he didn't have a representative at the drawing, didn't know what slots the airline got and didn't know yet what it would do with them. "Until we see what we got and see where it fits in with the rest of our geographical operation, I can't say," he explained.

Some industry sources said the carrier probably could make a significant amount of money by trading their attractive slots--as landing rights are called--for some less attractive ones.

The Miami-based Fleming has federal authority to carry passengers nationwide, but currently operates only cargo service, overnight, to 13 U.S. cities. The nine-year-old airline once operated scheduled passenger service between Worcester, Mass., and Puerto Rico but got out of it because of fare wars, Fleming said. "With the Braniff situation, there are some routes out there that might be lucrative," he said. The carrier's fleet consists of 3 Boeing 727s, 1 McDonnell Douglas DC8 and 8 Lockheed Electras.

"We're a low-keyed airline," Fleming said. "We keep things kind of quiet down here."

An FAA spokesman said yesterday that Fleming may not be eligible for the National slots because the government's policy on National requires the slots to go to a carrier operating planes with 56 or more seats. Besides the two Washington airports, Fleming got first crack at Braniff slots at Seattle, New Orleans, Newark, Corpus Christi and Harlingen, Tex. It also got the second spot for slots at several other airports.

In the drawing, New York Air was second at National and Frontier Airlines was third. Each is eligible for two slots. A spokesman for Frontier said it would be trading the National slots for slots elsewhere.

The drawing was held to determine the priority in which airlines would be eligible for 255 slots previously allocated to Braniff, which suspended service two weeks ago.

The Braniff slots being handed out through the random drawing are good for 60 days, subject to recall by the FAA on 24-hour notice. "If Braniff does again operate, then the slots necessary for continued Braniff operations will be returned to Braniff," the FAA said.

Altogether, 76 airlines put their names in for the drawing. The slots can be traded among the airlines, but only for other Braniff slots.