Permutation
Statistics & ProbabilityA 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.
Example
How many ways can $3$ students finish first, second, and third in a race if there are $5$ students? $P(5,3) = 5 \times 4 \times 3 = 60$ ways. Order matters because $1$st, $2$nd, $3$rd are different places.
Key Insight
Use permutations when ORDER matters. The letters A, B, C in the order A-B-C is different from B-A-C.
Definition
A permutation of $n$ items taken $r$ at a time is an ordered selection: $P(n,r) = n!/(n-r)!$. The total number of permutations of $n$ distinct items is $n!$. Permutations count the number of ways to arrange $r$ items from $n$, where changing the order creates a different permutation.
Example
How many ways can a $4$-digit PIN be created using the digits $1$-$9$ with no repetition? $P(9,4) = 9!/5! = 9 \times 8 \times 7 \times 6 = 3{,}024$ possible PINs.
Key Insight
The key question for permutations vs. combinations: does order matter? Permutation = order matters. Combination = order does not matter. ABC and CBA are the same combination but different permutations.
Definition
The number of permutations of $n$ distinct objects is $n!$. The number of $r$-permutations from $n$ objects is $P(n,r) = n!/(n-r)!$. 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. This is the multinomial coefficient.
Example
The word MISSISSIPPI has $11$ letters: $M(1)$, $I(4)$, $S(4)$, $P(2)$. Number of distinct arrangements $= 11!/(1! \cdot 4! \cdot 4! \cdot 2!) = 39916800/1152 = 34{,}650$.
Key Insight
Permutation groups (symmetric groups $S_n$) are fundamental in abstract algebra. The symmetric group $S_3$ has $6$ elements (all permutations of $3$ objects) and is the smallest non-abelian group, connecting combinatorics to group theory and physics (particle symmetries).