If there are 7 people in a room, but only 4 chairs..

by JK Thu Sep 11, 2008 8:48 pm

On page 57 of the manhattangmat guide for word translation problems, I have a question on the seating arrangements problem. why does order matter for the 4 seats?

If there are 7 people in a room, but only 4 chairs available, how many different seating arrangements are possible?

The answer is 7!/3!

I just don't get why seating matter matters...can't 4 chosen people sit in any of the 4 chairs? So order does not matter?

by **JPG** Fri Sep 12, 2008 10:44 am

Did you ever get an answer to this one?

by JK Fri Sep 12, 2008 3:53 pm

no but i figure order matters because they are seated, whereas if they were just put on a team or on a side of the room, order does not matter. in this case there is only one seat avail for 4 people so order matters.

by **StaceyKoprince** Thu Oct 09, 2008 5:54 pm

yep, order matters on this one, because when we're discussing "seating arrangements" then the actual arrangement of the individual seats matters!

As opposed to, say, being chosen for a team (or not). There, it's a yes/no proposition - either yes, you're on, or no, you're not on. If they had said you either get a seat or you don't get a seat, that would also be a yes/no proposition. But a seating *arrangement* = who sits in which specific seat. In that case, there's a different between: Amy-Mary-Joe-John and Amy-Mary-John-Joe.

by **shriek4me** Fri Oct 24, 2008 7:10 am

1st chair has 7 options 2nd chair has 6 options 3rd chair has 5 options 4th chair has 4 options hence 7*6*5*4 which can be written as 7!/3!

by **RonPurewal** Sat Nov 15, 2008 7:05 am

JK Wrote:I just don't get why seating matter matters...can't 4 chosen people sit in any of the 4 chairs? So order does not matter?

this is the reason why i don't particularly like the phrase "order matters" for describing this particular issue.
instead of thinking about whether "order" matters, you should instead consider the following question: if you interchange any of the elements/people in a chosen set, do you get a new set that's different from the original?
if the answer is yes, then "order matters"; if the answer is no, then "order doesn't matter".

in this case, if you take the same 4 people but switch around their seats, then you definitely have a different arrangement. therefore, you must consider this problem as though "order matters".

see here for another problem on which i give a description of this issue.