How many different groupings possible?

Expert replies

Vemuri: Legendary Member; Posts: 682; Joined: Fri Jan 16, 2009 2:40 amFollowed by: 1 member; Thanked: 32 times

by Vemuri » Sun Mar 22, 2009 5:24 am

Coach Miller 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 forwaard. 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. 2,580
D. 3,360
E. 151,200

Like this post

Source: Beat The GMAT — Problem Solving | Sun Mar 22, 2009 5:24 am

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

Re: How many different groupings possible?

by Brent@GMATPrepNow » Sun Mar 22, 2009 7:24 am

Vemuri wrote:Coach Miller 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 forwaard. 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. 2,580
D. 3,360
E. 151,200

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)

Brent Hanneson - Creator of GMATPrepNow.com

Like this post

Vemuri: Legendary Member; Posts: 682; Joined: Fri Jan 16, 2009 2:40 amFollowed by: 1 member; Thanked: 32 times

Re: How many different groupings possible?

by Vemuri » Mon Mar 23, 2009 12:56 am

Brent Hanneson wrote: 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)

Spot on !!!

Brent, how would you solve the problem if order matters in the defence & forward?

Like this post

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Mar 23, 2009 7:57 am

Brent, how would you solve the problem if order matters in the defence & forward?

Since order would then matter, those stages would require permutations:
Stage 2 would be 8P2 ways (56)
Stage 3 would be 6P2 ways (30)

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

Like this post

Vemuri: Legendary Member; Posts: 682; Joined: Fri Jan 16, 2009 2:40 amFollowed by: 1 member; Thanked: 32 times

by Vemuri » Mon Mar 23, 2009 9:42 am

Thanks Brent.

Like this post

vishaljainsxc: Junior | Next Rank: 30 Posts; Posts: 16; Joined: Sun Jan 30, 2011 6:34 amFollowed by: 1 member

by vishaljainsxc » Wed Sep 28, 2011 11:57 pm

why is the answer not 2C1 x 8C5..since out of the 8 ppl we have to select 3 people

Like this post

studentps2011: Senior | Next Rank: 100 Posts; Posts: 40; Joined: Fri Jul 15, 2011 5:37 am; Thanked: 3 times

by studentps2011 » Thu Sep 29, 2011 9:40 am

vishaljainsxc wrote:why is the answer not 2C1 x 8C5..since out of the 8 ppl we have to select 3 people

We can select 5 people from 8 (8C5) to get 56 groups. But then, the 5 boys in each group can be arranged in (5!/(2!x2!) = 30)ways.

So total number of groupings = 2x56x30 = 3360

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

559 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review