Lottery Basics:
So there's 14 ping pong balls numbered 1-14, and 4 balls are selected at random. There are 1001 possible combinations, but the league only assigns out an even 1000.
Now we'll have the teams ranked by worst records:
#1 (worst team) - 250 combinations
#2 (2nd worst team) - 199 combinations
.
.
.
#14 (best team not in playoffs) - 5 combinations
Combinations are assigned to teams in sequential order (with the order the number is drawn in not mattering so 1,2,3,4 is the same as 4,3,2,1). So the team in the #1 spot gets the first 250 combinations:
1,2,3,4
1,2,3,5
1,2,3,6
.
.
.
1,7,11,14
1,7,12,13
1,7,12,14 (what I believe is the 250th combination).
And this continues on and on with the team in the #14 spot getting the last 5 combinations of:
10,11,12,13
10,11,12,14
10,11,13,14
10,12,13,14
(with the league not giving out the 1001st combination of 11,12,13,14):
So whoever the rep is sitting in the room for the #1 worst team, they know that pretty much if the #1 is pulled along with any number 7 or less, they've won. The #14 team knows that it has to be all numbers 10 or higher for them to win. It's not like you're looking through 250 random lottery tickets trying to match up the numbers.
So in your hypothetical, it's very easy to know which slot the assigned number came from and which team it belonged to. Was it combination 1-250, 251-449, 450-605, etc.?
Now if a team has multiple combination ranges like in the OPs hypothetical with Boston having the ping pong ball combinations for the teams in the #1 (1-250), #2 (251-449), and #3 (450-605) spots. If one of the first 605 combinations gets pulled, Boston wins, but it still depends on which combination was pulled as the balls have to be redrawn for picks 2 and 3. So if combination #147 gets pulled Boston wins, they'll then put all the balls back and redraw again for pick #2, now if combination #218 gets drawn, Boston doesn't win the #2 pick (not yet anyways), because they already won with one of the first 250 combinations, so they'll have to keep redrawing until a combination outside of #1-250 is drawn. If combination #604 gets drawn for the #2 pick, yay Boston wins again this time from the third slot, so they'll put the balls back and redraw again and they'll keep redrawing until a combination outside of 1-250 and 450-605 gets drawn.
Hopefully that makes sense.