Combinatorics Basics
Permutations and combinations are part of a branch of mathematics called combinatorics, which involves studying finite, discrete structures. They help determine the number of possible outcomes in various situations.
Order Matters!
The primary difference between a permutation and a combination is whether the order of selection is important. In a permutation, the order matters (e.g., a combination lock should technically be called a permutation lock, because 1-2-3 is different from 3-2-1). In a combination, order does not matter (e.g., choosing a committee of 3 people from a group).
Variables
- nTotal number of elements
- rNumber of elements selected
Permutations (nPr)
Permutations without replacement refer to all the possible ways elements can be listed in a particular order, where an element cannot be chosen more than once. The generalized equation is:
For example, choosing a President and a Vice-President from a group of 11 people: 11! / (11 - 2)! = 11! / 9! = 11 × 10 = 110 ways.
Combinations (nCr)
Combinations are related to permutations but remove redundancies, since order is not important. You divide the number of permutations by the number of ways the selected elements can be arranged among themselves (r!). The generalized equation is:
For example, choosing 2 members for a committee from a group of 11 people: 11! / [2! × 9!] = 55 ways.