The following Question:

Greg, Marcia, Peter, Jan, Bobby and Cindy go to a movie and sit next to each other in 6 adjacent seats in the front row of the theatre. If Marcia and Jan will not sit next to each other, in how many different arrangements

Answer will be posted tomorrow

total ways of arranging 6 ppl = 6! = 6*5*4*3*2 = 720

ways in which m&j sit together = 5!*2 = 240

ans = 720 - 240 = 480

@krishp84


We cannot apply symmetry because the number of cases where M and J are sitting together is not equal to the number of cases where M and J are not sitting together.
Consider 3 people: A, B and C
They can be arranged in 3! = 6 ways
ABC
ACB
BAC
BCA
CAB
CBA
In how many of these are A and B sitting together? 2
In how many are they not sitting together? 4
When we arrange 6 people here in 6! (= 720) ways, in how many of those will M and J sit together?
Consider M and J to be one and arrange 5 people in 5! ways. Also, M and J can exchange places so multiply by 2.
You get 5!*2 = 240 ways
Out of 720, M and J will be next to each other in only 240 ways.



That question would be something like this: 6 people go to a movie and sit next to each other in 6 adjacent seats in the front row of the theatre. If Marcia will not sit to the right of Jan, how many different arrangements are possible?

'to the right of Jan' means anywhere on the right, not necessarily on the adjacent seat. Here we see symmetry because there are only 2 ways in which Marcia can sit. She can sit either to the left of Jan (any seat on the left) or to the right of Jan (any seat on the right). There is nothing else possible. The number of cases in which she will sit to the left will be same as the number of cases in which she will sit to the right. That is why the answer here will be 6!/2.

But the original question talks about sitting right next to each other on adjacent seats. The probability of sitting 'next to each other' is less than the probability of sitting 'not next to each other'. Hence we cannot apply the symmetry principle there.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save 10% on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

Variant-2.1 of this question -
7 people(A,B,C,D,E,F,H) go to a movie and sit next to each other in 7 adjacent seats in the front row of the theatre. A will not sit to the left of F and F will not sit to the left of E in how many different arrangements ?
_________________

Labor cost for typing this post >= Labor cost for pushing the Kudos Button
kudos-what-are-they-and-why-we-have-them-94812.html

Variant-4.1 of this question -

7 people(A,B,C,D,E,F,H) go to a movie and sit next to each other in 8 adjacent seats in the front row of the theatre. A will not sit to the left of F and F will not sit to the left of E in how many different arrangements ?
_________________

Labor cost for typing this post >= Labor cost for pushing the Kudos Button
kudos-what-are-they-and-why-we-have-them-94812.html

Variant-3 of this question -
7 people(A,B,C,D,E,F,H) go to a movie and sit next to each other in 8 adjacent seats in the front row of the theatre. A and F will not sit next to each other in how many different arrangements ?

-->

8P7-2[(8-2+1)P(7-2+1)] = 8P7-2(7P6) ways = 8! - 2(7!) = 6(7!) ways

The answer is correct. An alternative approach could be this:

I will just think of the vacant seat as a person V.
So 8 people, 8 seats in 8! ways.
2 people should not be together so 8! - 2*7!

Another interesting variant could be this:
7 people(A,B,C,D,E,F,H) go to a movie and sit next to each other in 8 adjacent seats in the front row of the theatre. In how many different arrangements will there be at least one person between A and F?

Try it.

Variant-4 of this question -
7 people(A,B,C,D,E,F,H) go to a movie and sit next to each other in 8 adjacent seats in the front row of the theatre. A ,F and E will not sit next to each other in how many different arrangements ?

Correct solution. Alternative approach parallel to the one above:

The vacant spot is a person V.
Required arrangements: 8! - 3!*6!
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save 10% on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

WoW - Loved this Approach --> Kudos +1


This is equivalent to saying 8 people sit in 8 adjacent seats with an imaginary girl Ms. X with them
Subtract the number of arrangements when Ms. X sits between A and F
AXF
FXA
XAF
XFA
AFX
FAX
All the above are equivalent - No person sits between A and F
So answer will be 8!-(3!)(6!) ways

Now let me apply my traditional approach and confirm this -
8P7-(3!)((8-3+1)P(7-3+1)) = 8P7-(3!)(6P5) = 8!-(3!)(6!) ways
_________________

Labor cost for typing this post >= Labor cost for pushing the Kudos Button
kudos-what-are-they-and-why-we-have-them-94812.html

This will be 8!/2 ways
or
(8P7)/2 = 8!/2 waysThis will be (1/2)(8!/2) = 8!/4 ways
or
(1/2)(1/2)(8P7) = 8!/4 ways
Logic - combination of variant 2.1 and 3.1I understand this is a HUGE post - But could not find another way to link all these related concepts.
Welcome to any new variant. I LOVE these puzzles.
_________________

Labor cost for typing this post >= Labor cost for pushing the Kudos Button
kudos-what-are-they-and-why-we-have-them-94812.html

Your solutions are correct.
In a few weeks, I am going to start Permutation Combination on my blog. I already intend to make a post on this question and all its variants.
If you want to try out further complications, try putting in 2 or 3 vacant spots. Now there will be 2-3 identical people named 'V' so adjustments will be needed.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save 10% on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

Yes - I was thinking on these lines....I will post some variants on this once I am free.

Also thought of combining Probability with this because it is so related especially - the symmetry part.
_________________

Labor cost for typing this post >= Labor cost for pushing the Kudos Button
kudos-what-are-they-and-why-we-have-them-94812.html