Return to Discrete
Unit 2.3

ART OF
COUNTING

Combinatorics is the study of finite discrete structures. It is not just about counting; it is about knowing how many possibilities exist without having to list them all.

PROBABILITY SPACE
To crack the code, you must
know the magnitude.

The Possibility Engine

Does order matter? This is the fundamental question. A Permutation is a lock (1-2-3 is unique). A Combination is a handshake (Me shaking your hand is the same as you shaking mine).

Security Mainframe

The Vault

Total Items (n) 5
Slots to Fill (k) 3
Example: A password. '1-2-3' is distinct from '3-2-1'.
P(5, 3) = 5! / (5-3)!
POSSIBLE OUTCOMES
60

The Language of Chance

n!
Factorial

The product of an integer and all the integers below it. (e.g., 5! = 5×4×3×2×1 = 120).

nPr
Permutation

An arrangement of items where ORDER MATTERS. (e.g., A password "1234" is different from "4321").

nCr
Combination

A selection of items where ORDER DOES NOT MATTER. (e.g., A fruit salad with Apple & Banana is the same as Banana & Apple).

⌈N/k⌉
Pigeonhole Principle

If n items are put into m containers, with n > m, then at least one container must contain more than one item.