# 6. Control questions and tasks / Контрольні запитання і завдання

## Counting principles

1. State the **product rule** and the **sum rule**. Which word — "and" or "or" —
   signals each?
2. A choice of $k$ objects from $n$ is classified by two questions. **Which two?**
   Name the formula in each of the four cases.
3. When does **order matter**? Contrast Задача 1 (plates) with Задача 4 (balls in
   boxes): why is one $n^k$ and the other a multiset?

## Arrangements, combinations, multisets

4. Write $P(n,k)$ and $C(n,k)$ and explain the relation $P(n,k)=C(n,k)\cdot k!$ in
   words.
5. How many three-letter "words" can be formed from an alphabet of $8$ letters (a) if
   letters may repeat, (b) if they may not?
6. Explain the **stars-and-bars** picture: why does distributing $k$ identical balls
   into $n$ boxes give $\binom{n+k-1}{k}$?
7. In how many ways can a committee of $3$ be chosen from $10$ people? How does the
   answer change if the three take **distinct roles** (chair, secretary, treasurer)?

## Probability

8. State the **classical (Laplace) definition** of probability. What must be true of
   the outcomes for it to apply?
9. State the **complement rule** and explain why it is the natural tool for
   "**at least one**" problems (as in Задача 9).
10. When are two events **independent**, and what is $P(A\cap B)$ then? Why are the two
    draws in Задача 6 independent but the two draws in Задача 7 **not**?
11. Define **conditional probability** $P(A\mid B)$. Show how Задача 8 is the
    multiplication rule $P(\text{good})\,P(\text{1st}\mid\text{good})$.
12. A box has $6$ white and $4$ black balls; two are drawn without replacement. Find
    the probability that (a) both are white, (b) at least one is black.