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# nPr Formula

The letter "P" in the ^{n}Pr formula stands for "permutation" which means "arrangement". ^{n}Pr formula gives the number of ways of selecting and arranging r things from the given n things. Sometimes the arrangement really matters. For example, if we have to find all the 3 digit numbers using the digits 1, 2, and 3, we would say the numbers to be 123, 132, 231, 213, 312, and 321. In this situation, the order of the digits matters to form different numbers. Let us learn the ^{n}Pr formula along with a few solved examples.

## What is nPr Formula?

^{n}Pr can be written as P (n, r) (or) ^{n} P _{r} (or) _{n} P _{r}. It is used to find the number of ways of selecting and arranging r different things from n different things. **nPr formula** is also known as permutations formula (as we call a way of choosing and arranging things to be a permutation). This formula involves factorials. Here is the ^{n}Pr formula.

^{n}Pr Formula

The ^{n}Pr formula is,

P (n, r) (or) ^{n} P _{r} (or) _{n} P _{r} = n! / (n - r)!

where

- n = total number of things
- r = number of things that have to be selected and arranged

## nPr Formula Derivation

Let us consider n different objects and assume that r different objects from them should be selected and arranged. Let us find a number of possible ways to do this.

- The number of ways of choosing the first object is n as there are n objects in total.
- Since the first object is already chosen, the number of ways of choosing the second object is (n - 1).
- Similarly, the number of ways of choosing the third object is (n - 2).
- While choosing the r
^{th}object, there are only (n - r + 1) objects are left and hence it can be chosen in (n - r + 1) ways.

By the fundamental counting principle, the number of ways (^{n}Pr) of selecting and arranging r different objects from n different objects is,

^{n} P _{r} = n (n - 1) (n - 2) ... (n - r + 1)

To simplify this, we use factorial notations. Multiplying and dividing the above expression by (n - r) ... 3 · 2 · 1,

^{n} P _{r} = [n (n - 1) (n - 2) ...(n - r + 1) (n - r) ... 3 · 2 · 1] / [(n - r) ... 3 · 2 · 1]

= n! / (n - r)!

Thus, the ^{n}Pr formula is derived.

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Let us see the applications of the ^{n}Pr formula in the section below.

## Examples Using nPr Formula

**Example 1: **Find the value of P(10, 4).

**Solution:**

Using nPr formula,

P(n, r) = n! / (n - r)!

Substitute n = 10 and r = 4 on both sides,

P(10, 4) = 10! / (10-4)!

= 10! / 6!

= 10 × 9 × 8 × 7 × 6! / 6!

= 5040

**Answer: **P(10, 4) = 5040.

**Example 2: **Find the number of 3 letter words that can be formed by rearranging the letters of the word MATH?

**Solution:**

The number of letters of the word MATH is n = 4.

The number of letters of each of the required words is r = 3.

Since the arrangement matters in the formation of the words, we apply the npr formula to find the required number of 3 letter words.

P(n, r) = n! / (n−r)!

P(4, 3) = 4! / (4−3) = 4! / 1!

= (4 × 3 × 2 × 1) / 1 = 24

**Answer: **The required number of 3 letter words = 24.

**Example 3: **8 students have participated in a running race competition and the top three students will be awarded the first, second, and third prizes. Find the number of ways in which the awarding can be done.

**Solution:**

The total number of students is n = 8.

The number of students who will be awarded = 3.

Since the arrangement among the first, second, and third prizes matters, we use the nPr formula to find the required number of ways.

P(n, r) = n! / (n−r)!

P(8, 3) = 8! / (8−3)! = 8! / 5!

= (8 × 7 × 6 × 5!) / (5!)

= 8 × 7 × 6

= 336

**Answer:** The possible number of ways = 336.

## FAQs on nPr Formula

### What is npr Formula?

The **npr formula** is used to find the number of ways in which r different things can be selected and arranged out of n different things. This is also known as the permutations formula. The ^{n}Pr formula is, P(n, r) = n! / (n−r)!.

### How to Derive nPr Formula?

We will find the number of ways of selecting and arranging r different objects from n different objects and we denote this by ^{n}Pr. Then the number of ways of selecting the first, second, third, ..., r^{th} object are n, (n - 1), (n - 2), ..., (n - r + 1) respectively. Then by the fundamental principle of counting, nPr = n (n - 1) (n - 2) ... (n - r + 1). By multiplying and dividing this by (n - r) ... 3 · 2 · 1, we get nPr = [n (n - 1) (n - 2) ...(n - r + 1) (n - r) ... 3 · 2 · 1] / [(n - r) ... 3 · 2 · 1]. This can be further written using factorials as, nPr = n! / (n−r)!.

### What are the Applications of^{ }npr Formula?

The nPr formula is used to find the number of arrangements (permutations) that can be made using r objects out of n objects. i.e., this formula can be used to find the number of words that can be formed by using a group of letters, the number of numbers that can be formed by using a group of digits, the number of ways in which a group of people can be arranged among themselves, etc.

### What is the Difference Between nCr Formula and nPr Formula?

The nCr formula is the combinations formula and is used to find the number of ways of selecting r things out of n things whereas the ^{n}Pr formula is the selection along with the arrangement of r things from n things. These formulas are:

^{n}Cr = n! / [(n - r)! r!]^{n}Pr = n! / (n−r)!.

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