Quarter Finals: How many different draws are possible?

[Touring Mars]

My mate and I have been struggling this afternoon trying to work out (for ourselves) just how many draws (of four games) can you get when 8 teams are put into a pot and draw out one by one...? I worked it out two different ways and got the answer both times, but I'm still not convinced that I'm right...

I say there are 105 different possible draws, meaning that you have 104-1 chance of guessing the actual draw. But others have gone as low as just 28-1.

Lil help?

 

[wfooshee]

28 combinations.

Formula is n!/(k!*(n-k)!) where n=8 (number of teams) and k=2 (how many are selected at a time) 2 items at a time out of 8.

For permutation the formula is n!/((n-k)!), but that doesn't apply here. Permutations treats team A vs team B as a different selection than team B vs team A, which it isn't. (Unless you want to include which team is home in the selection, in which case A[home] vs B is different from A vs B[home].) Then you have 56 possible permutations.

And yes, I looked it up.

I'm not sure, but I think your number is high because you re-used some teams somewhere in your table of possibilities. Team A can't start vs B and C, for example, filling 2 starting slots.

 

[Touring Mars]

Ha! I Googled this question again and I found this thread

28 is the number of unique pairs you can make with 8 teams...

BUT... what I was trying to find was the number of different combinations of 4 pairs you can make in s draw of 8 teams into four ties.

I still get 105 unique sets though, disregarding home and away etc.

Anyway, new question - if three teams are from the same country, what are the chances of at least one tie containing both teams?

I get 43%

Amirite?

 

[Famine]

Uhh... it depends on what you regard as unique.

28 is your any order (AB = BA), any position (AB in match 1 = AB in match 2, 3 or 4) group of pairs:
AB AC AD AE AF AG AH BC BD BE BF BG BH CD CE CF CG CH DE DF DG DH EF EG EH FG FH GH

If you regard each team drawn in each position as unique (A1B2 being distinct from B1A2), then the chance of you guessing the complete draw from first drawn team to last is 1 in 40,320. That's 1/8 * 1/7 * 1/6 *1/5 * 1/4 * 1/3 * 1/2 * 1 (because you'll get the last team right if you got all seven others )


If there's three teams from the same country - let's say they're Teams A, B and C - then there's six possible ties that put two of them together (AB, BA, AC, CA, BC, CB) out of 56 possible ties - 10.7%

 

[Touring Mars]

It's just the number of possible quarter final draws I'm trying to find...

e.g. Draw 1: AB CD EF GH; Draw 2: AB CD EG FH etc. Considering AB and BA as the same thing, I get 105 different possible draws...

If ABC are from the same nation, then I get 45 possible sets of those 105 that include either AB, AC or BC...

 

[Famine]

I'd presume that you'd take CD AB EF GH as being the same thing as AB CD EF GH?

Just bashing it out and thinking while I'm typing, where AB or BA is any one of the four ties, there are fifteen other possible sets of ties:
CD EF GH; CD EG FH; CD EH FG;
CE DF GH; CE DG FH; CE DH EF;
CF DE GH; CF DG EH; CF DH EF;
CG DE FH; CG DF EH; CG DH EF;
CH DE FG; CH DF EG; CH DG EF

The same would hold out for AC, AD, AE, AF, AG and AH - 105 ties as you say.

If A, B and C were the three teams from the same nation, the all fifteen of each of the AB and AC groups would be intranational (30) along with fifteen in total (all BC ties) from the AD, AE, AF, AG and AH groups - 45 ties as you say.

 

[Touring Mars]

Yay, someone in 2008 owes me a fiver ....