Examples on nPr 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.

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.