This module teaches how to count arrangements, selections, and possibilities using permutations, combinations, binomial coefficients, and inclusion-exclusion. It covers probability fundamentals, conditional probability, random variables, and iconic problems like the Monty Hall dilemma to handle uncertainty. These tools are crucial for analyzing algorithm efficiency, game design, randomized systems, machine learning, and risk assessment.

Counting techniques provide systematic methods for determining the number of possible outcomes in discrete structures. This topic introduces basic counting principles such as the sum rule and product rule.

This topic studies methods for counting arrangements and selections of objects. It distinguishes between ordered and unordered selections and introduces formulas for permutations and combinations.

Binomial coefficients arise in counting combinations and in the expansion of binomial expressions. This topic covers the binomial theorem, Pascal’s identity, and important combinatorial identities.

The inclusion–exclusion principle provides a systematic way to count elements in overlapping sets. It is widely used in counting problems involving unions of multiple sets.

This topic introduces probability as a measure of uncertainty based on counting outcomes. It defines experiments, sample spaces, events, and basic probability rules.

Conditional probability measures the likelihood of events given prior information. This topic introduces independence and Bayes’ theorem, enabling probabilistic reasoning in real-world decision making.

Random variables assign numerical values to outcomes of random experiments. This topic covers discrete and continuous distributions, expectation, and variance, forming the foundation of probability modeling.