What is nPr Formula? | How to Use nPr Formula with Examples (2024)

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. The nPr formula is, P(n, r) = n! / (n−r)!, and is also called Permutation Formula.

In this article, we learn about nPr formula, its significance, properties, mathematical derivation, and diverse applications across mathematics and real-world scenarios.

What is ⁿP_r Formula?

A permutation is an arrangement of all or part of a set of objects, about the order of the arrangement. The nPr formula is used to calculate the number of permutations of n distinct objects taken r at a time. It is denoted mathematically as:

ⁿP_r = n! / (r!(n – r)!)

where,

• “n” is the total number of items in the set,
• “r” is the number of items to be chosen, and
• “!” denotes factorial, which is the product of all positive integers from 1 to the given number.

nPr Formula

nPr formula is used when we have to choose “r” options out of “n” choices. And the nPr formula is,

P (n, r) = ⁿP_r = _nP_r = n! / (n – r)!

Where,

• n is Total Number of Things
• r is Number of Things that have to be Selected and Arranged

Properties of nPr Formula

Some of the common properties of the ⁿP_r Formula are:

• ⁿP_n = ⁿP_n-1

Proof:

LHS: ⁿP_n = n!/(n-n)! = n!/0! = n!

RHS: ⁿP_n-1 = n!/[n-(n-1)]! = n!/1! = n!

Thus, LHS = RHS

• ⁿP_r = n × ^n-1P_r-1

Proof:

LHS: ⁿP_r = n!/(n-r)!

RHS: n × ^n-1P_r-1 = n × (n-1)!/[(n-1)-(r-1)]! = [n × (n-1)!]/(n-r)! = n!/(n-r)!

Thus, LHS = RHS

• ⁿP_r = ^n-1P_r + ^{r × (n-1)}P_r-1

Proof:

LHS: ⁿP_r = n!/(n-r)!

^n-1P_r = (n-1)!/(n-r-1)!

^{r × (n-1)}P_r-1= r × (n-1)!/[(n-1)-(r-1)]! = r × (n-1)!/(n-r)!

RHS: ^n-1P_r + ^{r × (n-1)}P_r-1 = (n-1)!/(n-r-1)! + r × (n-1)!/(n-r)! = (n-1)!(n-r)/(n-r)! + r × (n-1)!/(n-r)!

⇒^n-1P_r + ^{r × (n-1)}P_r-1= (n-1)!(n-r+r)/(n-r)! = (n-1)!n/(n-r)! = n!/(n-r)!

Thus, LHS = RHS

Derivation of ⁿP_r Formula

Let the n different objects be a₁, a₂, a₃, . . . , a_n.

• First place can be filled up by any one of the n objects in n ways.
• Second place can be filled up by any one of the remaining (n-1) objects in (n-1) ways.
• Third place can be filled up by any one of the remaining (n-2) objects in (n-2) ways.

This continues and goes on till the r^th place is filled.

Number of ways of filling up the rth place = n-r+1

Using Fundamental Counting Principle,

Total permutations (nPr) = n × (n – 1) × (n – 2) × … × (n – r + 1) = (n!)/(n-r)!

This formula calculates the number of ways to arrange ‘r’ elements out of ‘n’ distinct elements without repetition.

ⁿP_r and ⁿC_r Formula

Combination means selection. Here, the order does not matter. Whereas permutation of n different objects taken r at a time = (number of ways of selecting r objects from n different objects) *× (number of ways of arranging the selected r objects) i.e., permutation is the way to arrange some objects.

Where,

• n is the total number of objects,
• r is the number of objects taken at a time, and
• Factorial i.e., n! is the product of all positive integers from 1 to “n.”

Relationship between ⁿP_r and ⁿC_r Formula

As ⁿP_r = (number of ways of selecting r objects from n different objects) × r!

⇒ (number of ways of selecting r objects from n different objects) = ⁿP_r /r!

⇒ ⁿC_r = ⁿP_r /r! = n!/(n-r)!(r!)

Thus, the relationship between ⁿP_r and nCr Formula is:

ⁿC_r = ⁿP_r /r!

Applications of Permutation (ⁿP_r) Formula

Various applications of ⁿP_r Formula are:

Combinatorial Analysis: This is about counting different ways things can be arranged.

For example, arranging people in a line, picking a group, or creating passwords with unique characters.

Probability and Statistics: In chance and data analysis, we use arrangements to figure out how likely certain outcomes are. This helps in understanding and predicting probabilities, which is important in statistics.

Generating Arrangements: In computers and encryption, arrangements are used to create unique sequences. This is handy for securing information, shuffling data, or making things random.

Genetics and Biology: In genetics, arrangements are studied to understand gene sequences and variations. This is crucial for genetic research and understanding how living things are put together at a molecular level.

Game Theory: In games and puzzles, arrangements are important for figuring out all the possible moves or solutions. This helps in planning strategies and solving problems when playing board games or puzzles.

Also learn, Application of Permulatation and Combination

Article Related to nPr Formula:

• Permutation
• Combinations
• Permutation and Combination

Examples on ⁿP_r Formula

Example 1: Suppose you have a deck of 52 playing cards, and you want to find the number of ways to choose 5 cards in a specific order from the deck (i.e., permutations of 5 cards out of 52).

Solution:

Here, n = 52 (total number of cards in the deck) and r = 5 (number of cards to be chosen).

⁵²P₅ = 52!/(52-5)!

⇒ ⁵²P₅ = 52!/(47)!

⇒ ⁵²P₅ = (52 × 51 × 50 × 49 × 48 × 47!)/47!

⇒ ⁵²P₅ = 311,875,200

Therefore, there are 311,875,200 different ways to choose 5 cards in a specific order from a standard deck of 52 playing cards.

Example 2: Seven athletes are participating in a race. In how many ways can the first 3 athletes win the prize?

Solution:

Here, n = 7 (total number of athletes participating in the race) and r = 3 (number of athletes to be chosen to win the prize).

⁷P₃ = 7!/(7-3)!

⇒ ⁷P₃ = 7!/(4)!

⇒ ⁷P₃ = 7 × 6 × 5 × 4!/4!

⇒ ⁷P₃ = 7 × 6 × 5 = 210

Hence, there are 210 ways 3 athletes can win the prize.

Example 3: In how many ways can 6 persons stand in a queue?

Solution:

Here, n = 6 (total number of persons) and r = 3 (number of persons to stand in the queue).

⁶P₆ = 6!/(6-6)!

⇒ ⁶P₆ = 6!/(0)! = 6!/1 = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Hence, there are 720 ways 6 persons can stand in a queue.

Example 4: A student has 12 different books on a shelf. In how many ways can the student arrange 7 books on the top shelf?

Solution:

Here, n = 12 (total number of books) and r = 7 (number of books to be arranged).

¹²P₅ = 12!/(12-5)!

⇒ ¹²P₅ = 12!/(7)!

⇒ ¹²P₅ = 12 × 11 × 10 × 9 × 8 × 7!/7!

⇒ ¹²P₅ = 12 × 11 × 10 × 9 × 8 = 95040

Hence, there are 95040 ways to arrange 7 books on a shelf out of 12 different books.

Practice Problems on ⁿP_r Formula

P1. If ²ⁿ⁺¹P_n-1:^2n-1P_n = 3:5, then find the value of n.

P2. How many different signals can be given using any number of flags from 5 flags of different colors?

P3. Find the sum of all the numbers that can be formed with the digits 2, 3, 4, and 5 taken all at a time.

P4. If ⁿP₅ = 20×ⁿP₃, find the value of n.

P5. How many numbers can be formed from the digits 1, 2, 3, and 4 when repetition is not allowed?

P6. How many 4-letter words, with or without meaning can be formed using the letters in the word LOGARITHMS, if repetition of letters is not allowed?

ⁿP_r Formula: FAQs

What are Permutations?

Permutations are arrangements of objects in a specific order. In mathematics, particularly in combinatorics, a permutation of a set is an ordered arrangement of its distinct elements.

What is ⁿP_r Formula in Probability?

The ⁿP_r formula is:

ⁿP_r = n! / (n – r)!

What is the Value of ⁿP₀?

ⁿP₀ is defined as 1. In combinatorics, the number of ways to arrange 0 elements (which essentially means having an empty set) is considered to be 1.

What is the Meaning of r in ⁿP_r?

In ⁿP_r, r is the number of objects chosen and arranged from the total n objects.

How do you know when to use ⁿC_r or ⁿP_r?

We use ⁿP_r when order of arrangement matters, whereas we use ⁿC_r when it doesn’t.

Is ⁿP_r always greater than ⁿC_r?

No, ⁿP_r is not always greater than ⁿC_r for same value of r and n.

What is the nPr and nCr formula?

The nPr and nCr formula are:

• ⁿP_r = n! / (n – r)!
• ⁿC_r = = n! / r! × (n – r)!