There were 13 of them feeling their oats last night, out of 1,200 applicants for the coveted White House Fellowships among the country's most select chosen few.

"I got the best one!" Secretary of Agriculture John R. Block called over his shoulder bringing particularly luminous smiles to two proprietary faces, the parents of his new special assistant, David L. Bere, 30, a Quaker Oats executive in Chicago.

FBI Director William Webster was equally enthusiastic about his new special assistant, W. Stephen Harty, 31, an advertising executive from New York.

"Advertising's a great background for us," said Webster. "We always need to try to sharpen our communications."

It was the annual debut of White House fellows, by tradition a close-up look at the Washington officialdom with whom they will be hobnobbing during the coming year. The reception was at Blair House where national security adviser William P. Clark was the official White House greeter. But earlier, across the street, it had been President Reagan who received the fellows and their families in the Cabinet Room.

"It's terrific to be in the center of where the action is," said Elaine L. Chao, 30, of Harrison, N.Y., a Citibank lending officer who will work in the White House office of policy development.

Across the drawing room, James Bere of Chicago, chairman of Borg-Warner, confided about his son, David: "He's coming in with a high degree of idealism to solve all that hunger in the Third World. But I don't want to discourage him because it's important."

In another part of the room were the Rev. Lewis Simon and his wife, Lavonne, of Mobile, Ala.

"I always urged him to try and never to say can't. That's not in my vocabulary," said Lavonne Simon, keeping an eye on her son, Kenneth, 29, an attorney who has been assigned to the attorney general's staff.

Joseph R. Lupica, a trial attorney from West Hartford, Conn., said the fellows had some say in where they were assigned. "There was sort of a mating dance that was done in the manner of a fraternity rush," said Lupica. "There were about four or five departments I was interested in, and the same four or five were interested in me."

Depending on their qualifications, fellows can be hired with as high a rating as a GS15, step 3, paying about $50,000 a year. The program runs for a year, and includes a trip abroad as well as a domestic trip.

One of the smallest classes since Lyndon Johnson established the program in 1964, the 1983-84 group boasts some "firsts." At 23, David A. Neuman of Los Angeles, who did research for television producer Norman Lear, is the youngest fellow. He's in the Office of Cabinet Affairs at the White House.

At 6-foot-7, Muliufi F. Hannemann of Honolulu is the tallest fellow in the program's history. He is also its first of Samoan/American ancestry.