Combinations problem - Need expert help - The Beat The GMAT Forum

Combinations problem - Need expert help

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voodoo_child: Master | Next Rank: 500 Posts; Posts: 405; Joined: Thu Feb 10, 2011 1:44 amFollowed by: 1 member; Thanked: 3 times

by voodoo_child » Wed Aug 01, 2012 11:57 am

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?

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Source: Beat The GMAT — Problem Solving | Wed Aug 01, 2012 11:57 am

tisrar02: Master | Next Rank: 500 Posts; Posts: 130; Joined: Fri Apr 20, 2012 8:13 amFollowed by: 4 members; Location: Toronto, Ontario; Thanked: 16 times; GMAT Score:650

by tisrar02 » Wed Aug 01, 2012 5:37 pm

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.

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GmatMathPro: GMAT Instructor; Posts: 349; Joined: Wed Sep 28, 2011 3:38 pmFollowed by: 54 members; Location: Austin, TX; Thanked: 236 times; GMAT Score:770

by GmatMathPro » Wed Aug 01, 2012 7:03 pm

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.

Pete Ackley
GMAT Math Pro
Free Online Tutoring Trial

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GMAT/MBA Expert

Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Thu Aug 02, 2012 9:19 pm

No. of ways to select 1 goalkeeper from 2 boys = 2C1
No. of ways to select 2 defense from 8 boys = 8C2
No. of ways to select 2 midfield from 6 boys = 6C2(because 2 boys can play goalkeeper only and 2 are already selected for defense, 10 - 2 - 2 = 6)
No. of ways to select 1 forward from 4 boys = 4C1 (because 2 boys can play goalkeeper only, 4 are already selected for defense and midfield, 10 - 2 - 4 = 4)

Total no. of selections = 2C1 * 8C2 * 6C2 * 4C1 = 3,360

The correct answer is D.

Anurag Mairal, Ph.D., MBA
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Gurome, Inc.
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