There were 105 (7 x 5 x 3 x 1) possible pairings of the 8 teams.
[7 is the possible number of pairings for the first team drawn out of the hat. 5 is then the possible number of pairings for the third team drawn out of the hat. 3 is then the possible number of pairings for the 5th team drawn out of the hat. And 1 is the possible number of pairings for the 7th team drawn out of the hat.]
The simplest way to understand all of this is as follows:
Take one English team, say Liverpool
The chances of Liverpool being drawn against teams other than Manure or Chelski were the chances of being drawn against 5 of the other 7 teams. That is 5 chances out of 7 (5/7).
Then take a second English team, say Manure.
The chances of Manure ALSO being drawn against a non-English team are therefore 4 chances out of the now 5 other teams (Chelsea being the only English team in those 5 other teams) (4/5).
The chances of both Liverpool and Manure being drawn against non-English teams therefore equals 5/7 x 4/5 = 4/7.
That is 4 chances out of 7.
Chelski's chances of ALSO being drawn against a non-English team would then be 100% (i.e 3 chances out of 3). And there is no additional impact on the probabilities.
So there were 4 chances out of 7 that the English teams would NOT be drawn against each other.
[7 is the possible number of pairings for the first team drawn out of the hat. 5 is then the possible number of pairings for the third team drawn out of the hat. 3 is then the possible number of pairings for the 5th team drawn out of the hat. And 1 is the possible number of pairings for the 7th team drawn out of the hat.]
The simplest way to understand all of this is as follows:
Take one English team, say Liverpool
The chances of Liverpool being drawn against teams other than Manure or Chelski were the chances of being drawn against 5 of the other 7 teams. That is 5 chances out of 7 (5/7).
Then take a second English team, say Manure.
The chances of Manure ALSO being drawn against a non-English team are therefore 4 chances out of the now 5 other teams (Chelsea being the only English team in those 5 other teams) (4/5).
The chances of both Liverpool and Manure being drawn against non-English teams therefore equals 5/7 x 4/5 = 4/7.
That is 4 chances out of 7.
Chelski's chances of ALSO being drawn against a non-English team would then be 100% (i.e 3 chances out of 3). And there is no additional impact on the probabilities.
So there were 4 chances out of 7 that the English teams would NOT be drawn against each other.
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