Problem Solving

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


OA is D

Here's what I did : 8C5 * 2C1 = 112.

OE - 2C1*8C2*6C2*4C1=3360
Can someone please explain why we need to split the combinations above? It shouldnt matter because defence, midfield and forward will be chosen from 8 boys....

Thoughts?

Once you account for a position, you have the remaining boys left in the pool.

OE - 2C1*8C2*6C2*4C1=3360
You took 2 boys for goalie
You have 8 boys to choose for 2 defense positions.
Now you have chosen those 2 boys, they are out of the pool!
You have 6 boys to choose for 2 midfield positions.
Now you have 4 boys left to choose from.
Then you take the last boy for the forward position from the last group of 4 which is why the OE is as it is.

Once the person is chosen, they are out of the pool indefinitely and cannot be replaced in questions like these which is why the combinations are split apart.

Hope this helps.

voodoo_child wrote:
Here's what I did : 8C5 * 2C1 = 112.

OE - 2C1*8C2*6C2*4C1=3360
Can someone please explain why we need to split the combinations above? It shouldnt matter because defence, midfield and forward will be chosen from 8 boys....

Thoughts?

8C5 would tell you how many ways you could choose 5 players from a group of 8, but it doesn't count all the different ways a unique group of five players could be distributed among the different positions. For example, A,B playing defense and C,D playing midfield is different from A,C playing defense and B,D playing midfield. To get the right answer, you need to not only count how many ways you can choose the 5 players, but how many ways you could put those five players in the different positions.