rocdocmac Posted June 13, 2019 Report Share Posted June 13, 2019 Suppose there is an international football (soccer) team consisting of 20 players and selected as follows from five countries: Two players from Spain Three from Italy Four from France Five from Brazil Six from Germany The squad only includes one goalkeeper, who plays for an Italian club, whereas the captain plays for a French club. Assuming the goalkeeper and the captain are included in every selection of eleven players, how many different teams could be selected from the twenty players (irrespective of position) if at least three German club players are included in each selection? Quote Link to comment Share on other sites More sharing options...
1 bonanova Posted June 17, 2019 Report Share Posted June 17, 2019 Spoiler I miscalculated a combination and transposed a 9 and 7.@Thalia fixed the combination and trustingly? kept my transposition. One more try ... Spoiler The Selection pool comprises 20 players: { SS III FFFF BBBBB GGGGGG } Removing the captain and goalkeeper the pool shrinks to 18: { SS II FFF BBBBB GGGGGG } Without the three-German restriction, we have 18 choose 9 = 48620 different teams. We must include {3 4 5 6} Germans, then choose {6 5 4 3} from the 12 non-Germans. (6 choose 3 = 20) x (12 choose 6 = 924) = 18480(6 choose 4 = 15) x (12 choose 5 = 792) = 11880(6 choose 5 = 6) x (12 choose 4 = 495) = 2970(6 choose 6 = 1) x (12 choose 3 = 220) = 220--------------------------------------------TOTAL 33550 different teams Quote Link to comment Share on other sites More sharing options...
1 Thalia Posted June 17, 2019 Report Share Posted June 17, 2019 (edited) Possibly both typos! (6C3 = 20) x (12C6 = 924) = 18,480 (6C4 = 15) x (12C5 = 792) = 11,880 (6C5 = 6) x (12C4 = 495) = 2,790 (6C6 = 1) x (12C3 = 220) = 220 33,370 Edited June 17, 2019 by Thalia Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted June 16, 2019 Report Share Posted June 16, 2019 Spoiler The Selection pool comprises 20 players: { SS III FFFF BBBBB GGGGGG } Removing the captain and goalkeeper the pool shrinks to 18: { SS II FFF BBBBB GGGGGG } Without the three-German restriction, we have 18 choose 9 = 48620 different teams. We must include {3 4 5 6} Germans, then choose {6 5 4 3} from the 12 non-Germans. (6 choose 3 = 20) x (12 choose 6 = 924) = 18380(6 choose 4 = 12) x (12 choose 5 = 792) = 9504(6 choose 5 = 6) x (12 choose 4 = 495) = 2790(6 choose 6 = 1) x (12 choose 3 = 220) = 220--------------------------------------------TOTAL 30994 different teams Quote Link to comment Share on other sites More sharing options...
0 rocdocmac Posted June 16, 2019 Author Report Share Posted June 16, 2019 Spoiler Almost there ... two mistakes in calculation, one of which is certainly a typo! Quote Link to comment Share on other sites More sharing options...
0 rocdocmac Posted June 17, 2019 Author Report Share Posted June 17, 2019 (edited) Spoiler Something is still wrong (third typo) ... the total is incorrect! Edited June 17, 2019 by rocdocmac Quote Link to comment Share on other sites More sharing options...
0 Thalia Posted June 18, 2019 Report Share Posted June 18, 2019 I see... I have a tendency to switch numbers in my head. All the right digits were there! Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted June 19, 2019 Report Share Posted June 19, 2019 Yes, we conquered it as a team effort. RDM, very nice puzzle. Quote Link to comment Share on other sites More sharing options...
0 rocdocmac Posted June 20, 2019 Author Report Share Posted June 20, 2019 (edited) Spoiler Yes, that's it!: 33550 Kudos to Thalia for spotting the typos! Edited June 20, 2019 by rocdocmac Quote Link to comment Share on other sites More sharing options...
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