Problem Solving

You have 2 choices for the goalkeeper. Because two of the ten don't play any other position, you have 8 choices left for the forward. I'd prefer it if the question said so explicitly (see below) but because of the question setup, I'll assume the order of the two defenders and of the two midfielders does not matter. Then we have 7*6/2 = 21 ways to choose the two midfielders, and finally 5*4/2 = 10 ways to choose the two defenders. So the answer is 2*8*21*10 = 3360.

I don't know much about soccer, but I assume there's a left defender and a right defender, and if so, the order of the two defenders would matter, which is why I have some reservations about the wording of the question.

Hi imskpwr,

There's a discussion of this question here:

https://www.beatthegmat.com/how-many-dif ... 33260.html

GMAT assassins aren't born, they're made,
Rich

imskpwr wrote:I think the answer to this Question is Wrong?

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

Divide the complete task into 4 stages:
1. Select a goalkeeper. We must select one boy from 2. We can accomplish this in 2 ways.
2. Select 2 for defence. We must select 2 boys from the remaining 8. We can accomplish this in 8C2 ways (28 ways.)
(Note: I'm assuming that order doesn't matter here. That is, there is no left defence and right defence; they are simply on defence)
3. Select 2 for midfield. We must select 2 boys from the remaining 6 boys. We can accomplish this in 6C2 ways (15 ways.)
4. Select 1 for forward. We must select 1 boy from the remaining 4 boys. We can accomplish this in (4 ways.)
The total number of ways to complete the entire task is 2x28x15x4 = 3360 (D)

imskpwr wrote:
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

We are given that there are 10 boys available to fill the following positions:

1 goalkeeper, 2 defenders, 2 midfielders, and 1 forward.

We are also given that only 2 of the boys can play goalkeeper.

Thus, we can select the goalkeeper in 2C1 = 2 ways.

We now have 8 boys left and need to select 2 defenders from those 8 boys:

8C2 = (8 x 7)/2! = 28 ways

We now have 6 boys left and need to select 2 midfielders:

6C2 = (6 x 5)/2! = 15 ways

We now have 4 boys left and need to select 1 forward:

4C1 = 4

Thus, the number of ways to select 6 starters is:

2 x 28 x 15 x 4 = 3,360

Answer: D