# Practical Work 6 — Combinatorics and Probability

> Sixth practical class of **Computer Discrete Mathematics**. Working **by hand**,
> you **count** the outcomes of a choice (arrangements, combinations, multisets) and
> use those counts to find **probabilities** — of independent draws, of conditional
> events, and of "at least one" via the complement.

## At a glance

| | |
|---|---|
| **Topic** | The two counting principles; arrangements $P(n,k)$ and tuples $n^k$; combinations $C(n,k)$ and multisets; classical (Laplace) probability; independence, conditional probability, and the complement rule |
| **Prerequisite lectures** | [L13 — Probabilities](../../Lectures/CDM-L13.md) (counting and combinatorics, classical probability, conditional probability, independence) |
| **Mode** | In-class problem solving — pen and paper, no computer |
| **Deliverable** | Worked solutions to the tasks in [3classroom.md](3classroom.md) |

## Contents

| # | Part | File |
|:--:|---|---|
| 1 | Objective | [1purpose.md](1purpose.md) |
| 2 | Methodical guidelines (self-contained theory) | [2method.md](2method.md) |
| 3 | Classroom tasks **with solutions** | [3classroom.md](3classroom.md) |
| 6 | Control questions | [6questions.md](6questions.md) |

## Conventions

- **Language** — all work is written in **English**.
- **By hand.** Set up each count or probability from a formula, show the substitution,
  and give the number; leave very large results as a product **and** its value.
- **Notation is fixed and self-contained.** $n!$ is the factorial; $P(n,k)=n!/(n-k)!$
  is the number of **arrangements** ($k$-permutations); $C(n,k)=\binom{n}{k}$ is the
  number of **combinations**; $P(A)$ (upright, with an event inside) is a
  **probability**. All are defined in [2method.md](2method.md).
- **Probabilities** are given as an exact fraction and a decimal (e.g.
  $\tfrac{1}{9}\approx0.111$).
- **Every task has a full solution.** [3classroom.md](3classroom.md) gives each task
  together with a complete worked answer.

## Summary

Combinatorics answers *how many?* and probability answers *how likely?* — and the
second is built on the first. Every count reduces to two ideas: **multiply** choices
made in sequence (the product rule) and decide whether **order matters** and whether
**repetition** is allowed, which selects one of four formulas — arrangements $P(n,k)$,
tuples $n^k$, combinations $C(n,k)$, or multisets $C(n{+}k{-}1,k)$. For equally-likely
outcomes, **probability is just counting**: favourable over total. Layer on three
rules — the **product** rule for independent trials, **conditional** probability when
one event informs another, and the **complement** ($P(A)=1-P(\overline A)$) for
"at least one" — and the nine tasks of this class are all within reach.