An interesting probability problem

Expert replies

billzhao: Senior | Next Rank: 100 Posts; Posts: 80; Joined: Mon Feb 02, 2009 6:36 am; Thanked: 10 times

by billzhao » Thu Feb 05, 2009 11:57 pm

Find the number of ways in which 4 boys and 4 girls can be seated alternatively in a row and there is a boy named John and a girl named Susan amongst the group who cannot be put in adjacent seats.

My thinking is: I put John and Susan together as (JS) or (SJ)
there are eight categories of combinations as below:

(JS)BGBGBG (SJ)GBGBGB
BG(JS)BGBG GB(SJ)GBGB
BGBG(JS)BG GBGB(SJ)GB
BGBGBG(JS) GBGBGB(SJ)

So the number of combination is: 2*4!*4! - 8*3!*3! Is it correct?

The answer is 2*4!*4! - 14*3!*3!

Thanks.

Yiliang

Source: Beat The GMAT — Problem Solving | Thu Feb 05, 2009 11:57 pm

fleurdelisse: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Tue Feb 03, 2009 2:01 am; Thanked: 1 times

by fleurdelisse » Fri Feb 06, 2009 1:44 am

You under-counted the number of times john and susan can be next to each other, it is 14. 7 for when you start the seating with a boy and 7 for when you start the seating with a girl.

you counted in pairs of (1,2), (3,4), etc. and did not account for susan and john to be in seats (2,3) for example. Do the counting again, and you'll see what I mean.

Reasoning for the whole answer, for the other readers:

Total number of ways that 4 boys an 4 girls can be seated alternatively is:
2*4!*4!, since you have 4! for seating girls and 4! for seating boys, and you multiply by 2 to account for the fact that the first seat can be either a boy or a girl

and then you remove the possibilities of having John and Susan together, which is in 14 different ways (count only for when a boy is in the first seat for example, you'll get 7 and then multiply by 2). And for each of the 14 different ways, you can seat the remaining girls and boys in 3!*3! ways

So end result: 2*4!*4! - 14*3!*3!

fleurdelisse: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Tue Feb 03, 2009 2:01 am; Thanked: 1 times

by fleurdelisse » Fri Feb 06, 2009 1:47 am

so, following your method of counting, it would be like this (in bold are those you did not account for):

(JS)BGBGBG (SJ)GBGBGB
B(SJ)GBGBG G(JS)BGBGB
BG(JS)BGBG GB(SJ)GBGB
BGB(SJ)GBG GBG(JS)BGB
etc.

billzhao: Senior | Next Rank: 100 Posts; Posts: 80; Joined: Mon Feb 02, 2009 6:36 am; Thanked: 10 times

by billzhao » Fri Feb 06, 2009 3:20 am

i got it. thanks a lot!

Yiliang

Uri: Master | Next Rank: 500 Posts; Posts: 229; Joined: Tue Jan 13, 2009 6:56 am; Thanked: 8 times; GMAT Score:700

by Uri » Fri Feb 06, 2009 3:23 pm

Is there any other way to find out the solution, without writing down the possible arrangements?

Featured schoolRound 2 closes Jan 7, 2027 54 days left

Chicago Booth

Since 1982, 10 Booth faculty members have received the Nobel Prize in Economic Sciences, more than any other business school.

#3
U.S. News rank: $175,000
Median base salary: 670
Average GMAT Focus: 635
Students in the class

Apply to Booth Visit chicagobooth.edu See the full profile

Photo: Warren LeMay, CC BY-SA 2.0

GMAT Course Reviews

563 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (563 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