Final answer:
Combinatorics includes four main classes for counting selections: ordered and non-repeated (permutations), unordered and non-repeated (combinations), ordered with repetition, and unordered with repetition. For permutations, the formula is n!, for combinations it is C(n, k) = n! / [k!(n-k)!], for ordered with repetition it is n^k, and for unordered with repetition, the formula is C(n + k - 1, k).
Step-by-step explanation:
Combinatorics Formulas
The study of counting, arrangements, and probability is known as combinatorics. There are four main classes for counting selections: ordered with repetition, unordered with repetition, ordered without repetition, and unordered without repetition.
Ordered and Non-Repeated (Permutations)
When dealing with ordered non-repeated selections, the formula is the factorial of the number of items. For example, the number of ways to arrange 4 distinct items is 4! (4 factorial), which is calculated as 4×3×2×1, or 24. This counts the number of permutations.
Unordered and Non-Repeated (Combinations)
For unordered non-repeated selections, we use the combination formula C(n, k) = n! / [k!(n-k)!], where n is the total number of items and k is the number being chosen.
Ordered with Repetition
When selections are ordered with repetition, the count is simply n^k, where n is the number of items to choose from and k is the number of selections being made.
Unordered with Repetition (Multiset Permutations)
For unordered with repetition, we utilize the multiset permutation formula C(n + k - 1, k), where k is the number of selections being made from n types of items.
Each of these formulas applies to different scenarios in combinatorics, allowing us to calculate probabilities, arrange items, and analyze various outcomes systematically.