Why is nPr called the number of permutations of n different objects taken r?

[s3a]

I'm just curious, why is nPr called the "number of permutations of n different objects taken r at a time"?

 

[phinds]

I'm just curious, why is nPr called the "number of permutations of n different objects taken r at a time"?

Well, because that's what it IS. What would you like to call it?

 

[s3a]

I meant that I don't understand what, in general, is being taken r at a time.

 

[SteamKing]

I meant that I don't understand what, in general, is being taken r at a time.

A set of r objects from a group of n total objects.

It could be playing cards, toppings you put on a pizza, whatever.

 

[phinds]

I meant that I don't understand what, in general, is being taken r at a time.

Whatever it is that you are taking the permutations of.

 

[s3a]

To be more specific, what is meant by "at a time"?

For example, the answer to the question "In how many ways can 5 differently coloured marbles be arranged in a row?" is nPn.

So let's visually illustrate this row of 5 marbles as follows.: _ _ _ _ _

In this case nPn = 5P5, which is read as the "number of permutations of 5 different objects taken 5 at a time". I get that there are 5 different objects, but I don't get what is being taken 5 at a time; each column will only have 1 marble, not 5.

P.S.
This may not have been the best example due to the fact that n = r, but I hope the point still came across.

 

[phinds]

Suppose you want 5 marbles taken 3 at a time. The marbles are red, black, white, green, yellow.

You can do
red, black, yellow
red, yellow, black
green, white, blue
.
.
.
and on and on, taking 3 marbles at a time out of your total of 5 marbles. The nPr is the total number of such permutations

 

[Ray Vickson]

To be more specific, what is meant by "at a time"?

For example, the answer to the question "In how many ways can 5 differently coloured marbles be arranged in a row?" is nPn.

So let's visually illustrate this row of 5 marbles as follows.: _ _ _ _ _

In this case nPn = 5P5, which is read as the "number of permutations of 5 different objects taken 5 at a time". I get that there are 5 different objects, but I don't get what is being taken 5 at a time; each column will only have 1 marble, not 5.

P.S.
This may not have been the best example due to the fact that n = r, but I hope the point still came across.

It means "(ordered) groups of r". I don't think the "at a time" part is particularly descriptive.

 

[s3a]

Suppose you want 5 marbles taken 3 at a time. The marbles are red, black, white, green, yellow.

You can do
red, black, yellow
red, yellow, black
green, white, blue
.
.
.
and on and on, taking 3 marbles at a time out of your total of 5 marbles. The nPr is the total number of such permutations

This was exactly what I was looking for!

It means "(ordered) groups of r". I don't think the "at a time" part is particularly descriptive.

This wasn't exactly what I was looking for, but it did help me understand something related that I read online.

Thank you both! :)

 

[haruspex]

It means "(ordered) groups of r". I don't think the "at a time" part is particularly descriptive.

I would say that the significance is that it denies replacement. On the other hand, it does tend to suggest no interest in the order, so it is a bit strange that it is used in the context of permutations.