Permutation
A permutation is an arrangement of objects in a specific order, where the order matters.
Formula
P(n, r) = \dfrac{n!}{(n-r)!}
Definition
A permutation is a way to arrange a group of things in a specific order; permutations count how many different ordered arrangements are possible. A permutation of $n$ items taken $r$ at a time is an ordered selection: $P(n,r) = n!/(n-r)!$, and the total number of permutations of $n$ distinct items is $n!$. Formally, for objects with repetition (multiset permutations), the count is $n!/(n_1!n_2!\cdots n_k!)$ where $n_i$ is the count of the $i$-th distinct object, the multinomial coefficient; permutation groups (symmetric groups $S_n$) are fundamental in abstract algebra, with $S_3$ (all permutations of $3$ objects) being the smallest non-abelian group.
Example
How many ways can $3$ students finish first, second, and third in a race of $5$ students? $P(5,3) = 5 \times 4 \times 3 = 60$ ways, since order matters (1st, 2nd, 3rd are different places). A $4$-digit PIN created from digits $1$-$9$ with no repetition has $P(9,4) = 9!/5! = 3{,}024$ possibilities. The word MISSISSIPPI has $11$ letters ($M(1)$, $I(4)$, $S(4)$, $P(2)$), so the number of distinct arrangements is $11!/(1! \cdot 4! \cdot 4! \cdot 2!) = 34{,}650$.
Key Insight
The key question for permutations vs. combinations is whether order matters: ABC and CBA are the same combination but different permutations. Permutation groups connect combinatorics to group theory and physics (particle symmetries), since $S_3$ has $6$ elements and is the smallest non-abelian group.