// assuming that Randomize was called somewhere at the application initialization

procedure TForm1.Button1Click(Sender: TObject);
var
  I: Integer;
  IntSum: Integer;
  IntArray: array[1..4] of Integer;
begin
  // initialize a helper variable which counts the sum of the first 3 random integers
  IntSum := 0;
  // iterate 3 times to generate first 3 numbers
  for I := 1 to 3 do
  begin
    // generate a random number in the range of 2 to 7
    IntArray[I] := Random(6) + 2;
    // and increment the helper sum variable by the just generated number
    IntSum := IntSum + IntArray[I];
  end;
  // and fill the last number, which must be 22 minus the sum of the previous 3 numbers,
  // which we have in the helper sum variable (so we don't need to calculate it anymore)
  IntArray[4] := 22 - IntSum;
  // now you should have in the IntArray array values between 2-7 whose sum is 22...
end;