$|\text{arrangements of } n \text{ objects}| = n! = n \times (n-1) \times \cdots \times 2 \times 1$ Combinatorics is the branch of discrete mathematics concerned with counting the number of ways to arrange, select, or partition objects, providing the tools needed to compute probabilities when outcomes are equally likely. The two fundamental problems are counting **ordered** arrangements (permutations) and **unordered** selections (combinations). In probability, combinatorics is used to determine the sizes of events and sample spaces, enabling calculation of $P(A) = |A| / |S|$ for uniform experiments. > [!example]- Counting Equally Likely Outcomes for a Lottery {💡 Example} > A lottery draws 3 numbers from $\{1, 2, 3, 4, 5\}$ without replacement. How many equally likely outcomes are there if order does not matter? > > > [!answer]- Answer > > This is a combination problem (order irrelevant, no replacement): > > $\binom{5}{3} = \frac{5!}{3!\,2!} = \frac{120}{6 \times 2} = 10$ > > There are 10 equally likely outcomes, so each has probability $1/10 = 0.10$.