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

• s3a
In summary, nPr is known as the "number of permutations of n different objects taken r at a time" because it represents the total number of ordered groups of r objects from a group of n total objects. This concept can also be applied to situations where the number of objects taken is less than the total number of objects, as it represents the number of possible arrangements without replacement.
s3a
I'm just curious, why is nPr called the "number of permutations of n different objects taken r at a time"?

s3a said:
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?

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

s3a said:
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.

s3a said:
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.

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.

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

s3a
s3a said:
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
phinds said:
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!

Ray Vickson said:
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! :)

Ray Vickson said:
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.

## 1. Why is nPr called "Permutation"?

nPr stands for "n Permutation" and is used to calculate the number of possible arrangements of a given set of objects. The term "permutation" refers to the act of arranging objects in a specific order or sequence.

## 2. How is nPr different from nCr?

While both nPr and nCr are used to calculate combinations and permutations, nPr is used when order matters, meaning the arrangement of the objects is important. nCr, on the other hand, is used when order does not matter, such as when selecting a committee from a group of people.

## 3. What is the formula for calculating nPr?

The formula for nPr is n!/(n-r)!, where n is the total number of objects and r is the number of objects being selected for the arrangement. The exclamation point represents the factorial function, which means multiplying a number by all the positive integers less than it.

## 4. Why is nPr important in mathematics and science?

nPr is important in mathematics and science because it helps in solving problems that involve arranging objects in a specific order. It is commonly used in probability, statistics, and combinatorics to calculate the number of possible outcomes in an experiment or event.

## 5. Can nPr be used in real-life situations?

Yes, nPr can be used in real-life situations, such as when calculating the number of possible outcomes in a game of cards or the number of ways to arrange a group of people in a line. It can also be used in business and marketing to calculate the number of possible combinations for a product or service.

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