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

Combinatorics #

1. Define a permutation, an arrangement, and a combination. Which of them depend on the order of the selected elements?
2. Write the formulas for $P_n$, $A_n^k$, and $C_n^k$. Derive $C_n^k$ from $A_n^k$.
3. Compute $P_5$, $A_5^3$, and $C_5^3$, showing the arithmetic.
4. What are $C_n^0$ and $C_n^n$? Explain why.

Descriptive statistics #

5. Define the mean (mathematical expectation), the variance, and the standard deviation of a dataset.
6. What is the difference between the population variance ($\div n$) and the sample variance ($\div (n-1)$)?
7. Why does the standard deviation, rather than the variance, have the same units as the data?

Implementation and complexity #

8. Why is the multiplicative form of $A_n^k$ preferable to computing $n! / (n-k)!$ directly?
9. Outline a recursive algorithm that generates all permutations of a set. What is its output size for $n$ elements?
10. Why is explicitly generating all permutations feasible only for small $n$, even though counting them is instant?

Tasks by hand #

11. Compute $P_4$, $A_6^2$, and $C_6^2$.
12. For the dataset $\{1, 3, 5, 7\}$, compute the mean, the (population) variance, and the standard deviation.
13. List all $C_4^2$ two-element combinations of the set $\{A, B, C, D\}$.

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