Properties of nPr Formula
Some of the common properties of the nPr Formula are:
- nPn = nPn-1
Proof:
LHS: nPn = n!/(n-n)! = n!/0! = n!
RHS: nPn-1 = n!/[n-(n-1)]! = n!/1! = n!
Thus, LHS = RHS
- nPr = n × n-1Pr-1
Proof:
LHS: nPr = n!/(n-r)!
RHS: n × n-1Pr-1 = n × (n-1)!/[(n-1)-(r-1)]! = [n × (n-1)!]/(n-r)! = n!/(n-r)!
Thus, LHS = RHS
- nPr = n-1Pr + r × (n-1)Pr-1
Proof:
LHS: nPr = n!/(n-r)!
n-1Pr = (n-1)!/(n-r-1)!
r × (n-1)Pr-1= r × (n-1)!/[(n-1)-(r-1)]! = r × (n-1)!/(n-r)!
RHS: n-1Pr + r × (n-1)Pr-1 = (n-1)!/(n-r-1)! + r × (n-1)!/(n-r)! = (n-1)!(n-r)/(n-r)! + r × (n-1)!/(n-r)!
⇒n-1Pr + r × (n-1)Pr-1= (n-1)!(n-r+r)/(n-r)! = (n-1)!n/(n-r)! = n!/(n-r)!
Thus, LHS = RHS
nPr Formula
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.
Table of Content
- What is nPr Formula?
- Properties of nPr Formula
- Derivation of nPr Formula
- nPr and nCr Formula
- Applications of Permutation (nPr) Formula
- Examples on nPr Formula
- Practice Problems on nPr Formula
- nPr Formula: FAQs