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I've just started looking over a new topic and am slightly confused on how to approach the following question..

A manager has 20 possible players to choose from for a Rugby Game. They are playing league, so need 13 players on the field and 4 on the bench giving a full team of 17.

From that, proceed to answer the following..

Suppose that the players can be given any position on the field. In how many ways can the coach select the full team?

Now I assumed it would be a case of going 17! but this gives an absurdly high answer which I feel is incorrect. I would appreciate if someone could lend a hand. Thanks !

Ergo
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  • first, choose the 13 players for the field, it's 20 choose 13, and then choose the other 4 from the remaining 7, it's 7 choose 4. totally it's the multiply of both – DanielY Sep 16 '13 at 06:42
  • If you are just choosing the $17$ players, without specifying who will play and who will be on the bench, the answer is $\binom{20}{17}$, which is $1140$. – André Nicolas Sep 16 '13 at 06:47
  • @AndréNicolas I think we need to make a distinction between bench players and field players. – DanielY Sep 16 '13 at 06:50
  • That may depend on the purpose of producing this group of $17$. League rules may specify that one can only "dress" $17$ players, and must supply the list to the referee. But typically in this kind of situation, the coach is free to use whever she wants as the starting team. – André Nicolas Sep 16 '13 at 06:56

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The answer can be calculated this way;

Choose the field players: $20\choose13$

Choose the bench players: $7\choose4$

Totally: $20\choose13$*$7\choose4$ = 2713200

DanielY
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  • Thanks a lot for your help! – Ergo Sep 16 '13 at 07:04
  • If I were to elaborate a step further with the following question -- Suppose that 10 of the players can only play “forward” positions and the other 10 can only play “back” positions. There are 6 forward positions and 7 back positions on the field. The coach aims for equal number of forwards and backs on the bench at the start of the game. In how many ways can the coach select the full team of 17 players (of forwards and backs)?

    Would the answer be (10C6) x (10C7) = 25,200?

     – Ergo Sep 16 '13 at 07:37
  • @Androx your answer would be only for the field players, not the bench – DanielY Sep 16 '13 at 08:19
  • @Androx i'm not sure I'm allowed to say it, but if you find my answer right and helpful, just click the small checkmark in the left of it, it'll be greatly appriciated :) – DanielY Sep 16 '13 at 08:22
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    No problem, cheers buddy. Isn't that what the question is asking for though? – Ergo Sep 16 '13 at 10:01
  • I didn't understand. what do you mean? – DanielY Sep 16 '13 at 10:03
  • The question in the comment above, is the answer I provided correct? Or do I need to manipulate the value 17 in somehow. – Ergo Sep 16 '13 at 10:31
  • it's true but only for the field players – DanielY Sep 16 '13 at 13:21