Permutations of n Distinct Objects
Simple Explanation
A permutation is an ORDERED arrangement. The number of different orders in which ALL n distinct objects can be arranged is simply n! — exactly the factorial just introduced.
Why Do We Need It?
This is the simplest, most direct application of factorials — arranging every object in a set, a starting point before the next concept generalizes to arranging only SOME of the objects.
Formula
Permutations of n Distinct Objects
P(n,n) = n!
The number of different orders in which all n distinct objects can be arranged.
- n
- — the total number of distinct objects being arranged
When to use it: Whenever every object in a set is being arranged in some order (not just a subset of them).
Worked Example
Count the arrangements of a full set
In how many different orders can 5 different books be arranged on a shelf?
Why Does This Work?
There are 5 choices for which book goes in the first position, 4 remaining choices for the second position, and so on down to exactly 1 choice for the last position — multiplying these choices together (Fundamental Counting Principle) gives 5×4×3×2×1 = 5!.
Real-Life Example
Arranging a race's finishing order
A race has 5 distinct runners, and organizers want to know how many different possible finishing orders exist.
5! gives every possible complete finishing order directly.
Practice
In how many different orders can 4 distinct trophies be arranged on a shelf?
MediumCommon mistake
Confusing arranging ALL n objects (n!) with arranging only SOME of them — a different, more general formula (covered next) is needed when only r out of n objects are being arranged.
Quick Review
- P(n,n) = n! — arranging every one of n distinct objects.
- Comes directly from the Fundamental Counting Principle, applied to each position in turn.
- A special case of the more general "permutations of r from n" formula, with r=n.