In UEFA Champions League, how is it ensured that two teams from the same Country aren't drawn into the same group?

Question

UEFA Champions League draw rules state that:

No team can play a club from their own association in the group stage and the first round of knockout phase. (source)

Since the teams are already placed in their pots, how is that rule ensured during the draw?

Are the balls marked somehow?

Answer 1

Shortly:

1. A team is drawn by a human (generally a famous player).
2. The computer indicates the possible groups in which that team can be drawn.
3. A human puts the corresponding balls of possible groups into a separate pot.
4. A group is drawn from that separate pot by a human (another famous player).

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More detailed:

There are 4 pots with the names of the teams allocated to these pots by their rankings and additional 8 pots each containing only the group letters (i.e, a pot with only "A"s, a pot with only "B"s and so up to a pot with only "H"s). Externally the balls that contain team or group names should look completely identical.

1. Firstly, teams from pot 1 are drawn (one team is drawn per each group without constraints).
2. For the draw with teams in the 2nd/3rd/4th pots there will be some constraints. So, after a team is drawn, the computer suggests in which groups that team may be allocated.
3. Then an attendant creates a separate pot with the only group-letters where the team can be drawn into. This operation is made after the draw of each team based on the most current constraints.
4. Finally, a group is drawn for the selected team from that separate pot.

This is the youtube video of 2015 Champions League draw.

Here you can see that Real Madrid is drawn. Computer shows that only groups A and D are available for Real Madrid and then the responsible takes one ball from pot A and another from pot D and puts them into a separate pot, from where the final post for Real Madrid is drawn.

At some point it's also possible that a team can be drawn only into one group (Valencia can be allocated only into group H), so there is no need to draw the group ball.

Answer 2

It is not that complicated. A total of 32 teams will compete in the UEFA Champions League. Which association will send how many teams is decided by the UEFA coefficients. Based on these numbers, associations like Spain, England and Germany could send four teams each and associations like Romania, Cyprus and Denmark, etc. could send only one team (based on the 2015-16 competition), those teams from associations with low coefficients should participate in First Qualifying Rounds, Second Qualifying Rounds, Third Qualifying Rounds.

After these rounds, a total of 22 teams:

12 champions from associations 1 to 12

6 runners-up from association 1 to 6

3 third-placed teams from association 1-3

Europa League title holders

+

10 winners from the third qualifying round will make 32 teams.

Therefore, the maximum number of teams that each association could send is only 5 when a title holder is not from the top four teams in the league.

There are four pots for drawing for each team.

Pot 1: Seven champions from the top seven countries + a title holder (winner of the last year's UEFA Champions League). If the title holder is from the 12 champions, there will be eight champions from the eight top countries.

Pot 2: Remaining five champions that didn't make it to Pot 1 + three runner-ups

Pot 3: Remaining three runner-ups + three third-placed teams + two qualifying round winners.

Pot 4: Eight remaining qualifying round winners.

There is another pot that has balls containing group names.

Drawing procedures are quite simple. For example, one ball will be drawn from Pot 1 (champions) and another ball will be drawn from the pot for group A to H to determine the group for the team. It is repeated until all the balls in Pot 4 are drawn. A group (which already has a team from the same country) can't be drawn for a team, because the ball contains the group will be eliminated before a drawing to ensure no team can play a club from their own association. For example, after ManU is drawn into Group A and Chelsea is drawn later, a ball that has Group A will not be in the pot.

A side note: For the 2015-16 Champions League, UEFA decided not to draw Russia and Ukraine into the same group based on the political situation between the two countries.