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# International Football Team

## Question

 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?

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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

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• 1

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 by Thalia
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• 0
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

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Almost there ... two mistakes in calculation, one of which is certainly a typo!

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Something is still wrong (third typo) ... the total is incorrect!

Edited by rocdocmac
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I see... I have a tendency to switch numbers in my head. All the right digits were there!

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Yes, we conquered it as a team effort.

RDM, very nice puzzle.

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Spoiler

Yes, that's it!: 33550

Kudos to Thalia for spotting the typos!

Edited by rocdocmac

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