# 1. Objective / Мета роботи

**Gain fluency in counting the outcomes of a choice — arrangements, combinations,
and multisets — and in turning those counts into probabilities of everyday
random experiments.**

By the end of the class the student should be able to:

- apply the **product rule** to a choice made in stages (car plates, lottery cards)
  and recognise when **repetition** is allowed ($n^k$) or not;
- tell **order-matters** from **order-ignored** problems and pick the right formula —
  an **arrangement** $P(n,k)=\dfrac{n!}{(n-k)!}$ when order matters, a **combination**
  $C(n,k)=\dbinom{n}{k}$ when it does not;
- count **distributions of identical objects** into distinct boxes with the
  **multiset** (stars-and-bars) formula $C(n{+}k{-}1,\,k)$;
- compute a **classical probability** as favourable outcomes over total, and combine
  events with the **product rule** for independent trials, the **conditional** rule
  when one event constrains another, and the **complement** rule
  $P(A)=1-P(\overline A)$ for "at least one";
- **model** an urn/selection problem from words to a formula, and give the answer as
  an exact fraction and a decimal.

The class consolidates
[Lecture 13 — Probabilities](../../Lectures/CDM-L13.md) (combinatorics
and the classical definition, conditional probability, and independence) and
complements [Lab 7](../../Laboratory/Laboratory7/main.md). All theory
needed is restated, self-contained, in [2method.md](2method.md).