Seating Arrangements in a Railway Compartment

[lionely]

Homework Statement

There are 8 seats in a railway compartment. In how many ways can 8 people be seated if 2 must have their backs to the engine and 1 must face the engine?

Homework Equations

The Attempt at a Solution

So 3 of them must be in those exact positions so I tried treating them like one entity so I'm ordering 6 things now, but I also need to order the 3 persons

so I have 6! * 3!, but this isn't the answer. What should I do?

 

[nejibanana]

I'm a little rusty with combinatorics, but if you ignore the stipulations, you have 8 seats. Each seat has two possible values, facing forward and facing backwards. So the total amount of combinations for that situation is 2*2*2... = 2^n = 2^8. With the stipulation, you remove 3 of the seats, as they only have one possible value now, so the total amount of combinations is 1*1*1*2*2*2..= 2^5.

 

[lionely]

i'm sorry but that's not the answer either :(

 

[Mark44]

All you need to be concerned about are the two people (of 8) who have to sit with their backs to the engine, and the one (of the remaining 6) who needs to sit facing the engine. The other five you don't need to worry about.

 

[lionely]

But isn't that what I was trying to do? :S The other 5 would just be 5! but the other three guys would 3! if I'm arranging them amongst themselves... I think...

 

[nejibanana]

I may have been thinking of permutation as opposed to combinations. Considering the permutations leads to undercounting I believe. For combinations, this page explains the situation a bit. http://en.wikipedia.org/wiki/Combination Basically it gives the formula for if you have n things and choose k of them. I think it will work for this situation, you have 8 things and you choose 3 of them.

 

[Mark44]

But isn't that what I was trying to do? :S

I don't think so. You said you were treating those three as a single entity, and this doesn't seem valid to me.

$$ {8}\choose{3}$$
is different from
$$ {{8}\choose{2}} \cdot {{6}\choose{1}}$$
The latter is a lot bigger.

The other 5 would just be 5! but the other three guys would 3! if I'm arranging them amongst themselves... I think...

 

[pasmith]

Homework Statement

There are 8 seats in a railway compartment. In how many ways can 8 people be seated if 2 must have their backs to the engine and 1 must face the engine?

Homework Equations

The Attempt at a Solution

So 3 of them must be in those exact positions so I tried treating them like one entity so I'm ordering 6 things now, but I also need to order the 3 persons

so I have 6! * 3!, but this isn't the answer. What should I do?

Given the constraints, you have the following possibilities:

Facing   Backwards   
1        7                
2        6
3        5
4        4
5        3
6        2

In each case there are then two questions: How many ways to choose those who sit facing the engine, and having done so, how many ways to order those who sit in the same direction.

 

[haruspex]

I don't think so. You said you were treating those three as a single entity, and this doesn't seem valid to me.

$$ {8}\choose{3}$$
is different from
$$ {{8}\choose{2}} \cdot {{6}\choose{1}}$$
The latter is a lot bigger.

I think you are supposed to assume there are four seats facing each way. Two of the people need to be seated in a particular four, and one in the other four.

 

[Mark44]

I think you are supposed to assume there are four seats facing each way. Two of the people need to be seated in a particular four, and one in the other four.

That's not an assumption I made. It's been a while since I rode on Amtrack, but my recollection is that some cars have seats that face only one direction (forward), and some, like the observation car, have seats in each of the four orientations. Also, there are lots of buses that have seat orientations other than just facing forward or backward.

 

[lionely]

So let me get this straight according to the question, the seats can only face towards to engine or the backwards to the engine?

 

[Mark44]

It's not stated in the problem (at least as far as your problem description goes). It's probably not an unreasonable assumption, but you should state it in your work.

 

[lionely]

Oh, but how would you work it without that assumption?

 

[Mark44]

Is there part of the problem you didn't include? Was there a diagram of the seat arrangement? If this is a problem your instructor wrote, you could ask for clarification on how the seats are arranged. I don't believe you can work the problem without having more information or making an assumption on the seat arrangement.

 

[lionely]

This is the exact question, it's from a book called Further Elementary Analysis by R.I. Porter, I found it on the internet. I could give you the link if you want it.

 

[haruspex]

This is the exact question, it's from a book called Further Elementary Analysis by R.I. Porter, I found it on the internet. I could give you the link if you want it.

If it's a British publication you can safely assume it's four seats forwards and four seats backwards.
Are the answers given?

 

[pasmith]

If it's a British publication you can safely assume it's four seats forwards and four seats backwards.

That would force there to be exactly four sitting in each orientation, which would make the given conditions on how many must sit in each orientation redundant.

 

[haruspex]

That would force there to be exactly four sitting in each orientation, which would make the given conditions on how many must sit in each orientation redundant.

I interpret the "must"s as being personal preferences. no redundancy. Again, insistence on facing a particular way may be a peculiarly British constraint.

 

[lionely]

It is a British publication and the answer is 5760

 

[haruspex]

It is a British publication and the answer is 5760

That answer fits my interpretation.