Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on th team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defense, 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?
Recognize that for the second slot, we only have 10-2 = 8 elements to choose from. We need 2 of 8 elements to fill that spot. 8C2
For the third slot, we only have 6 elements left to choose from. We need to fill it with 2 elements.
Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on th team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defense, 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 * 8C2*6C2*4C1
This is important to note, we have already chosen 2 goalkeeprs, so we are left with 8 people, and then after choosing 2 defense, we are left with 6 out of which to choose 2 midfield and then we are left with 4.
Hmm...but a good question... and it can be missed during the real exam! If proper attention is not paid!
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I cant get it. Why it is not C, but P here or your resulting calculation is random, pl give yr logic explaination!
N=8P5*2P1/(2P2*2P2)=3360 8P5 - we choose 5 boys of 8 (without goalkeepers) for 5 positions: 2 on defense, 2 in midfield, and 1 forward. 2P2*2P2 - we can change position within 2 on defense, 2 in midfield. So, we should exclude this variations. 2P1=2C1 - we choose goalkeeper of 2 boys
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thanks. that was from princeton but how do you know to break it up into: (8 C 2) * (6 C 2) * (4 C 1)
instead of just doing (8 C 5)
bmwhype2 I still have not understood why it can't be 8c5. Can you explain in detail?
If you consider 8C5, you are missing all the possible positions of the players. In a group of 5 players the coach can built many teams just switching the positions of those 5 players.
This problem has many approachs:
1st approach: n=2*C(8,5)*C(5,2)*C(3,2) = 3360; where you first consider two players for the goalkeeper position [2], second all the possible groups of 5 players from 8 players [C(8,5)] and third and last all the possible positions of all those 5 players [C(5,2)*C(3,2)].
2nd approach: n=2*8!/3!= 13440, number to which you have to discount all the permutations among defenses and midfields, i.e. 2! and 2!. Therefore 13440/[2!·2!]=3360
3th approach: n=2*C(8,2)*C(6,2)*C(4,1)=3360; you just has to consider how many players you can fill the positions with. Since there is no difference between mildfield1 and mildfield2 you "count" combinations. If not, you should count permutations.
why are selecting from 8 players whereas we will be left with 9 players after we made a selection for the goal keeper (as that is the logic stated in the solution for 6C2 i.e we are choosing from 6 because we will be left with 6 after selecting 2 from 8)
Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on th team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defense, 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?
Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on th team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defense, 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
Soln: 1 goal keeper can be chosen from 2 boys in 2C1 ways. 2 defenders can be chosen from 8 boys in 8C2 ways 2 midfielders can be chosen from left over 6 boys in 6C2 ways 1 forwards can be chosen from the left over 4 boys in 4C1 ways
Thus total number of ways of choosing team is = 2C1 * 8C2 * 6C2 * 4C1 = 3360
Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on th team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defense, 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
Goal Keeper selection = 2c1 Since only 2 can play at that position Defence Selection = 8c2 Midfield Selection = 6c2 Forward Selection = 4c1
Total combinations = 2c1 x 8c2 x 6c2 x 4c1 = 2 x 24 x 15 x 4 = 3360
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Re: Coach Miller is filling out the starting lineup for his [#permalink]
I agree with walker - this is a permutation problem. As selecting 1 goalkeeper from 2 will result in 2 different teams. Likewise since each remaining players can play all positions - though order will matter, yet we need to divide by 2! for each positions to ensure there are no repetitions. Please correct me if I went wrong in my understanding.
Bunuel
Re: Coach Miller is filling out the starting lineup for his [#permalink]
I agree with walker - this is a permutation problem. As selecting 1 goalkeeper from 2 will result in 2 different teams. Likewise since each remaining players can play all positions - though order will matter, yet we need to divide by 2! for each positions to ensure there are no repetitions. Please correct me if I went wrong in my understanding.
I'm not sure I understand you point about the order. Anyway below is a different approach to this problem:
Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on th team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defense, 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)
Hi Bunuel, I am not clear as to do this with P or C...though your answer with C matches mine with P. My explanation is same as Walkers. except for goalkeepers all other positions are common - should we not then divide by 2P2*2P2? Since selecting 2 goalkeepers was critical in terms of order - i used P.pls advice
Bunuel
Re: Coach Miller is filling out the starting lineup for his [#permalink]
Hi Bunuel, I am not clear as to do this with P or C...though your answer with C matches mine with P. My explanation is same as Walkers. except for goalkeepers all other positions are common - should we not then divide by 2P2*2P2? Since selecting 2 goalkeepers was critical in terms of order - i used P.pls advice
P and C just represent different formulas, different ways of counting. Most combinations questions can be solved in multiple ways, and if you understand the concept it really doesn't matter which approach you take.
As for this question: since we are dealing with different groups to be chosen from total and the order in each specific group doesn't matter I would use the method described in my previous post. It seems more straightforward and easy, at least for me.
Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on th team, and he must assign 6 starters to the following positions: 1 goalkeeper, 3 on defense, 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, without considering the order?
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