Permutation/Combination

Here is the problem:

In how many ways can seven airplanes line up for a departure on a runway if the plane with the greatest number of passengers must depart first?

Just want to know which formula to use..permutation or combination and how to use it [=

thanks!

MaeMarshmellow said:

Here is the problem:

In how many ways can seven airplanes line up for a departure on a runway if the plane with the greatest number of passengers must depart first?

Just want to know which formula to use..permutation or combination and how to use it [=

thanks!

Click to expand...

As I read the problem, if the # of passengers on each plane are different - there is only one way to line up.

Hello, MaeMarshmellow!

The problem is vaguely worded . . .

In how many ways can seven airplanes line up for a departure on a runway
if the plane with the greatest number of passengers must depart first?

Click to expand...

You want a formula?
. . Use the "Fullest-Plane-First" formula, if you can find it.
. . (Don't confuse it with the "Fullest-Taxi-First" formula.)


Assuming we know which plane has the most passengers (say, it's plane \(\displaystyle A\)),
. . it will be first in line.

Then the other six planes can be lined up in \(\displaystyle 6!\) ways.

Depends on the size of the runway in relation to the wing span of the planes.
If they can line up as in "lanes", as in athletics, then the plane with the largest number of passengers
can be in any "lane", if no other planes interfere with it's take off.
They could line up in 7! ways for such a configuration.
This allows for the possibility of a "parallel" line-up,
extremely unlikely with a commercial airline,
for which such a flight schedule is also impractical.


More practically, the situation may involve "serial" line-up, each plane coming out
onto a runway capable of accommodating a single plane.

If the plane on the runway with the largest number of passengers must always take off
first, each time a plane takes off, then it's been shown already that there is only one way this can happen,
given that each plane contains a different amount of passengers (do they?).

If the plane with the largest number of passengers must take off first,
subsequently then it is irrelevant how the remaining planes take off, that's also been answered.

When "order" matters, you use permutations or arrangements.
When order doesn't matter, you use combinations, which is in fact arrangements "unarranged".