With 18 teams at TI, the expected "average" amount of teams per region expected is 3 (direct + qualifier). See below for each region's likely scenario and best case scenarios based on major performances!
IG is guaranteed to qualify and, based on the current regional standings and remaining games, Aster, PSG LGD, and Vici are all very likely to qualify (good chance at ending up in the 700-800 point region). There will then be the 1 qualifier slot for Elephant, Ehome, and RNG to fight for. There is a strong likelyhood that China ends up with 5 slots for TI – with a worst case of 4 slots if regional placements are suboptimal with upsets and tie-breakers – even without any major points from the next major.
South America is nearly guaranteed 3 slots (2 direct invites) with Beast Coast and TP's likely point allocation – however, if Noping can get a top 8 performance and secure 1st in their region, they can make it 4 slots! Can you imagine the storyline if SA ends up with more slots at TI than EU?
CIS is a mixed bag. VP is guaranteed – however due to the decline of Na'vi in the 2nd season and the boost of Spirit, if neither Spirit or Na'vi perform at the major, CIS will only have 2 slots at TI (VP + qualifier). They will both need top 6 finishes to secure a slot. Likely to have 2 slots, chance at 3 slots, very low chance at 4 slots.
NA is guaranteed 2 slots (EG + qualifier). There is a moderate/decent chance of 3 slots (if QC can either take 1st place in region, or 2nd place + top 8 at major), and a very low chance at 4 slots (EG don't qualify to major and Undying take at least 1st seed or second seed + top 8 in addition to the above QC performance).
EU is currently effectively guaranteed 3 slots at TI (Secret, Alliance and Qualifier). For extra slots, Liquid/OG/Nigma need very high major performances, depending on their Season 2 finishes. Nigma have more room with 200 pts from first season, Liquid has less with 100 pts, and OG is in the worst spot with 42.5. If OG make it through as 4th place (with 100 points), even if they win the major they would have less than a 50% chance of qualifying directly. Nigma has a more realistic path forward – at least 2nd place DPC and top 8, 3rd place DPC and top 6, or 4th place DPC and top 3 leave them with 700 points, making it decently likely to qualify (similar story for Liquid, except 1 threshold higher for major placements). Overall, guaranteed 3 slots, with a low chance for 4 slots and an extremely low chance for 5 slots (just by nature of how these teams may knock each other out + large variance in these tourneys).
South East Asia
This is the region in the worst spot right now, as in they're most dependent on specific positioning within the league and major performances. If Fnatic somehow don't get top 3 after tie-breakers and SEA doesn't overperform in the major, they might actually have 0….that's right, 0 direct invites to TI, with only 1 slot from the regional qualifiers. However, if Fnatic can get at least 3rd place, they will guarantee at least 700 points, which is a likely qualification for TI (I think 60%+?). The issue with SEA is that the major point earner from the region, Neon, has a decent chance of not being in the top 4 in the region, meaning that they won't get enough points to qualify and won't have the opportunity to at the major. The region can have a best-case scenario though, if Fnatic are top 3 in the region, Neon make it to top 4, and TNC come in 1st in the region and make top 8 or 2nd in the region and top 3. Basically likely 2 spots, low chance at 3 spots, very low chance at 4 spots.
With all that said, keep a close eye on how the regions close out as there will be many cool storylines going into the major for TI invites. There's a very good chance that we'll see a 18-team TI allocation that we've never seen before in the history of DOTA!