Permutations (Order Matters)
A permutation is an arrangement of objects in a specific order. The number of permutations of n distinct objects taken r at a time is:
P(n, r) = n! / (n - r)!
Example: How many ways to arrange 3 books from 5?
P(5, 3) = 5! / (5-3)! = 5! / 2! = 60
Permutations of all n objects: P(n, n) = n!Combinations (Order Doesn't Matter)
A combination is a selection of objects where order doesn't matter.
C(n, r) = n! / (r! × (n - r)!) = n choose r
Example: How many ways to choose 3 books from 5?
C(5, 3) = 5! / (3! × 2!) = 10
Note: C(n, r) = P(n, r) / r!Permutations with Repetition
When objects can be repeated:
n^r ways to arrange r objects chosen from n types with repetition
Example: How many 3-digit numbers can be formed from digits 0-9?
10³ = 1000
Permutations of multiset:
n! / (n₁! × n₂! × ... × nₖ!)
Example: Arrangements of MISSISSIPPI:
11! / (1! × 4! × 4! × 2!) = 34650