AI Tutor
If your gym locker code is 44-11-99, typing 99-11-44 will leave you locked out, even though you pressed the exact same three numbers.
That is because the order of the items matters completely — and in mathematics, an ordered arrangement of objects is called a permutation.

Order Changes Everything

A selection (or combination) only cares who is in the group, while a permutation cares about the precise order or position of each member.
If selecting 33 items gives 11 group, but arranging them gives 66 distinct outcomes, how do we count permutations when the groups get much larger?

The Slot Method and Formula

Permutations rely on the Fundamental Counting Principle (or Multiplication Rule), which states that the total number of outcomes for sequential choices is found by multiplying the options available at each step.
Under the Fundamental Counting Principle (or Multiplication Rule), multiplying the choices at each step gives the total permutations: P(n,r)=n(n−1)(n−2)…(n−r+1)P(n, r) = n(n - 1)(n - 2)\dots(n - r + 1).
When filling rr positions from nn unique items, there are nn ways for the 1st position, n−1n - 1 for the 2nd, down to n−r+1n - r + 1 for the rr-th position, giving the product P(n,r)=n(n−1)(n−2)…(n−r+1)P(n, r) = n(n - 1)(n - 2)\dots(n - r + 1).
We write this arrangement count as nPr_n P_r, defined algebraically as nPr=n!(n−r)!_n P_r = \frac{n!}{(n-r)!}, where the factorial n!n! means multiplying all whole numbers from nn down to 11.
How do you instantly recognize whether a real-world scenario is asking for a permutation or just a simple selection?