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# A Coach is filling out the starting lineup for his indoor

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A Coach is filling out the starting lineup for his indoor [#permalink]  25 Oct 2009, 16:43
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A Coach is filling out the starting lineup for his indoor soccer team. There are 10 boys on the team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defence, 2 in midfield, and 1 forward. Only 2 of the boys can play goalkeeper, and they cannot play any other positions. The other boys can each play any of the other positions. How many different groupings are possible?

A. 60
B. 210
C. 2580
D. 3360
E. 151200
[Reveal] Spoiler: OA

Last edited by Bunuel on 04 Jul 2012, 00:39, edited 2 times in total.
Edited the question and added the OA
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  25 Oct 2009, 17:08
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Expert's post
rvthryet wrote:
A Coach is filling out the starting lineup for his indoor soccer team. There are 10 boys on the team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defence, 2 in midfield, and 1 forward. Only 2 of the boys can play goalkeeper, and they cannot play any other positions. The other boys can each play any of the other positions. How many different groupings are possible?

A) 60

B) 210

C) 2580

D) 3360

E) 151200

2C1 select 1 goalkeeper from 2 boys;
8C2 select 2 defense from 8 boys (as 2 boys can only play goalkeeper 10-2=8);
6C2 select 2 midfield from 6 boys (as 2 boys can only play goalkeeper and 2 we've already selected for defense 10-2-2=6);
4C1 select 1 forward from 4 boys (again as 2 boys can play only goalkeeper, 4 we've already selected for defense and midfield 10-2-4=4)

Total # of selection=2C1*8C2*6C2*4C1=3360

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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  25 Oct 2009, 17:57
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Small doubt.. Why should this not be 2C1 x 8C5??

I just can't seem to understand how is my thinking flawed there, although it is quite obvious that it is
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  25 Oct 2009, 18:38
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rvthryet wrote:
Small doubt.. Why should this not be 2C1 x 8C5??

I just can't seem to understand how is my thinking flawed there, although it is quite obvious that it is

Imagen different situation 4 players, we should choose 1 for defense and 1 for forward. (no restrictions).

The way you are doing you'll get 4C2=6. But look at the real case.

ABCD (players):

Defence - Forward
A B
A C
A D

B A
B C
B D

C A
C B
C D

D A
D B
D C

Total 12 possibilities 4C1*3C1=4*3=12. You just narrowed possible ways of selection.

In original question we are not choosing 5 people from 8, but we are choosing 2 from 8, than 2 from 6, than 1 from 4 (well and before we chose 1 from 2 as goalkeeper). And this is more ways of selection than 8C5 as you can see in the example.
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  25 Oct 2009, 18:55
rvthryet wrote:
Small doubt.. Why should this not be 2C1 x 8C5??

I just can't seem to understand how is my thinking flawed there, although it is quite obvious that it is

For that matter we could have gone an extra step and said it should be 9c6 (leaving other goalie out) , but for Bunuel explanation )
Bunuel is awesome.. of course this time I was clear too and I in fact prepared a similar example as Bunuel
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  26 Oct 2009, 06:12
2 goal keepers can be selected in 2 ways.

Rest 5 positions has to be filled from 8 boys.

2 Defence can be selected in 8C2 ways

2 Midfielder can be selected in 6C2 ways

1 forward can be selected in 4C1 ways.

So the total combinations 8*28*15*4 = 3360 ways
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  05 Feb 2011, 06:40
i made the same mistake as rvthryet... thanks for the explanation even tho i still dont understand how it makes such a different. i can c from your example that it happens. but i cannot understand the logic behind it. at the end he is choosing 5 ppl out of 8. the order have no effect.

but from what u are saying - unless its very clear that i need to choose only 5 ppl from a group without ANY distinction - it will be 5C8...

thanks bunuel.
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  12 Nov 2011, 02:17
Bunuel wrote:
rvthryet wrote:
A Coach is filling out the starting lineup for his indoor soccer team. There are 10 boys on the team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defence, 2 in midfield, and 1 forward. Only 2 of the boys can play goalkeeper, and they cannot play any other positions. The other boys can each play any of the other positions. How many different groupings are possible?

A) 60

B) 210

C) 2580

D) 3360

E) 151200

2C1 select 1 goalkeeper from 2 boys;
8C2 select 2 defense from 8 boys (as 2 boys can only play goalkeeper 10-2=8);
6C2 select 2 midfield from 6 boys (as 2 boys can only play goalkeeper and 2 we've already selected for defense 10-2-2=6);
4C1 select 1 forward from 4 boys (again as 2 boys can play only goalkeeper, 4 we've already selected for defense and midfield 10-2-4=4)

Total # of selection=2C1*8C2*6C2*4C1=3360

Sorry for bringing up an old post. I am clear with the above explanation except for one fact . Is there a necessity to choose in the above order. What happens if I choose the goalie, then the forward and then the defense & midfield. In such a case the combinations change drastically.

2C1 * 8C1 * 6C2 *4C2 = 1440.

The questions does not specify any order with selecting one group over another.
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  12 Nov 2011, 15:32
gmatrant wrote:
Sorry for bringing up an old post. I am clear with the above explanation except for one fact . Is there a necessity to choose in the above order. What happens if I choose the goalie, then the forward and then the defense & midfield. In such a case the combinations change drastically.

2C1 * 8C1 * 6C2 *4C2 = 1440.

The questions does not specify any order with selecting one group over another.

That would still work. Except that there is a mistake in the counting you did
2C1 * 8C1 * 7C2* 5C2 = 3360
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  04 Jan 2012, 22:00
Nice question. Easy to make the 8C5 trap. The individual selections need to be made and that, IMO, is the key to this problem. Answer is D.
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  02 Feb 2012, 20:30
Bunuel wrote:
rvthryet wrote:
A Coach is filling out the starting lineup for his indoor soccer team. There are 10 boys on the team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defence, 2 in midfield, and 1 forward. Only 2 of the boys can play goalkeeper, and they cannot play any other positions. The other boys can each play any of the other positions. How many different groupings are possible?

A) 60

B) 210

C) 2580

D) 3360

E) 151200

2C1 select 1 goalkeeper from 2 boys;
8C2 select 2 defense from 8 boys (as 2 boys can only play goalkeeper 10-2=8);
6C2 select 2 midfield from 6 boys (as 2 boys can only play goalkeeper and 2 we've already selected for defense 10-2-2=6);
4C1 select 1 forward from 4 boys (again as 2 boys can play only goalkeeper, 4 we've already selected for defense and midfield 10-2-4=4)

Total # of selection=2C1*8C2*6C2*4C1=3360

Bunnel for these type of question do we need to follow the positions as given in the question stem. Cant we first select 1 forward first and then the defence and midfield. If we go this way the no of selections will become 2c1*8c1*7c2*5c2. Lemme know your views on this
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  02 Feb 2012, 22:47
Expert's post
subhajeet wrote:
Bunnel for these type of question do we need to follow the positions as given in the question stem. Cant we first select 1 forward first and then the defence and midfield. If we go this way the no of selections will become 2c1*8c1*7c2*5c2. Lemme know your views on this

You can choose 2 in defense, 2 in midfield, and 1 forward from 8 in ANY order, you'l get the same result. The formula gives # of different selections of 2, 2, and 1 possible from 8, so the result must be the same despite the order in which you make this selections.
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  03 Jul 2012, 20:41
abhi47 wrote:
Guys,

Is it possible to solve this problem using the slot method ? Please clarify.

Yes it is possible to solve this problem using slot method: here is how I solved it

There are five slot's and ten players.

Most restricted slot is goal keeper: 2 players can go for that slot

Next is defender : out of 8 two can be selected : 8c2 = 28

Next is midfielder : out of remaining 6 two can be selected: 6c2 = 15

Next is forward : out of 4 1 can be selected : 4c1 = 4

Hence, total number of ways for selecting players = 2*28*15*4 = 3360

hope it is clear
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Re: A Coach is filling out the starting lineup for his indoor [#permalink]  05 Nov 2012, 20:07
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The formula can be simplified for slot method.

First is goalkeeper restriction: 2 options.
Next: Out of rest 8 players we need to fill slots with 2, 2, 1 and 3 (non-assigned) players.
Numbers of people in the same position are listed in denumeretor with factorial.

8! / (2! x 2! x 1! x 3!) = 1680
2 x 1680 = 3360.

As you can see the order of picking players doesnt matter.

Solution of Bunuel simply leads to the same formula:
$$2 * \frac{8!}{2!6!} * \frac{6!}{2!4!} * \frac{4!}{1!3!} = 2 * \frac{8!}{2!2!3!}$$
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Re: A Coach is filling out the starting lineup for his indoor [#permalink]  17 Nov 2012, 10:52
I did it in the next way:
1) ways to chose Goalkeepers - it's obvious 1C2=2
2) ways to chose Field players within the group - comes from DDMMF p=5!/2!2!=30
3) ways to chose Field players from 8 possible options - 5C8=56

Multiple everything: 2*30*56=3360

However the previous comment is the most elegant
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Re: Combination or Permutation: Can't make up my mind :) [#permalink]  24 Dec 2012, 01:06
144144 wrote:
i made the same mistake as rvthryet... thanks for the explanation even tho i still dont understand how it makes such a different. i can c from your example that it happens. but i cannot understand the logic behind it. at the end he is choosing 5 ppl out of 8. the order have no effect.

but from what u are saying - unless its very clear that i need to choose only 5 ppl from a group without ANY distinction - it will be 5C8...

thanks bunuel.

With the 8C5 logic, you are not accommodating the case where 2 defenders are different from 2 midfielders.

To pick up on the 4 player example Bunuel gave, 4 players - A,B,C,D we should choose 1 for defense and 1 for forward. (no restrictions).

When we do 4C2
Defence-forward
AB
AC
BC
BD
CD

But we are missing the case CA where C is defender and A is forward. In this particular example we can use the nPr permutation formula.

Back to original question, Bunuels method is pretty kickass.
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Re: A Coach is filling out the starting lineup for his indoor [#permalink]  28 Dec 2012, 03:45
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rvthryet wrote:
A Coach is filling out the starting lineup for his indoor soccer team. There are 10 boys on the team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defence, 2 in midfield, and 1 forward. Only 2 of the boys can play goalkeeper, and they cannot play any other positions. The other boys can each play any of the other positions. How many different groupings are possible?

A. 60
B. 210
C. 2580
D. 3360
E. 151200

How many ways to select goal keeper? 2 only, others cannot do it
How many ways to select 2 in midfield? $$=\frac{8!}{2!6!}=28$$
How many ways to select 2 on defence? $$=\frac{6!}{2!4!}=15$$
How many ways to select 1 in forward? $$=\frac{4!}{3!1!}=4$$

$$2*28*15*4 = 120*28 = 3360$$

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Re: A Coach is filling out the starting lineup for his indoor [#permalink]  03 Jan 2013, 22:56
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2C1 * 8C2 * 6C2 * 4C1
Ans - D
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Re: A Coach is filling out the starting lineup for his indoor [#permalink]  09 Feb 2013, 09:59
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I always preferred slot method to solve these problems since childhood.

But my following approach gave me a wrong answer.
2 8 7 6 5 4 =13440.

What i forgot to do is divide (8*7) with 2 and 6*5 with another 2(since order doesnt matter between 2 defenders and 2 midfielders).

One of the other way of doing this is.

$$2P1$$$$*8P5$$/2*2.
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Re: A Coach is filling out the starting lineup for his indoor [#permalink]  25 Oct 2013, 02:20
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Guys... since we have 6 positions in the field which are to be filled with 10 players.

Look at this: Filling the goal keeper position in 2 ways.
Next: the rest of the 5 positions are to be filled with 8 players and each player at a different position would be 8p5 But since we have 2 positions each for the defence and mid. divide it with 2! twice.

So answer is : 2* (8P5/(2!*2!)) = 3360

A very simple and easy approach based on the basics of P 'n C

Kudo me if you like this.
Re: A Coach is filling out the starting lineup for his indoor   [#permalink] 25 Oct 2013, 02:20

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