1. Objective / Мета роботи #

Implement the counting formulas of combinatorics and the basic descriptive statistics of a dataset, and generate combinatorial objects.

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

• compute the number of permutations $P_n = n!$, arrangements $A_n^k = \dfrac{n!}{(n-k)!}$, and combinations $C_n^k = \dfrac{n!}{k!\,(n-k)!}$, and explain when order matters;
• compute the mean (mathematical expectation), variance, and standard deviation of a set of values;
• generate and list the actual permutations, arrangements, or combinations of a given set, not merely count them;
• compute factorials and the counts efficiently, avoiding needless overflow;
• reason about the time complexity of counting versus generating.

The work provides the combinatorial groundwork for probability and prepares the tools used to reason quantitatively about finite sample spaces.

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