Combinatorics

Abstract

Combinatorics solves the problems of counting the number of various configurations (e.g., permutations), formed by the elements of finite sets, on which various restrictions can be imposed. The number of configurations formed by the elements of the set is counted in accordance with the sum rule and the product rule.

The sum rule states that if $A_1$, $A_2$, ..., $A_k$ are independent events, and there exists $n_1$ of possible outcomes of the event $A_1, n_2$ possible outcomes of the event $A_2$, ..., $n_k$ possible outcomes of the event $A_k$, then the possible number of outcomes of the event “$A_1~\mathbf{{or}}~A_2~\mathbf{{or}}\dots \mathbf{{or}}~A_k$” is equal to the sum $n_1+n_2+\cdots +n_k$.

The product rule states that if we have a sequence of k events with $n_1$ possible outcomes of the first one, $n_2$—of the second one, and so on, up to $n_k$ possible outcomes of the last one, then the total number of outcomes of the sequence of k events is equal to the product $n_1\cdot n_2\cdot \cdots \cdot n_k$.

$\left( n, k\right)$-Sample or sample of volume k out of n elements is any set of elements $x_1,x_2,\ldots , x_k$ out of the set A of cardinality n. And if the sequence of elements is of the essence, then the sample is called ordered, otherwise—unordered. An ordered $\left( n, k\right)$-sample with repeated elements is called $\left( n, k\right)$-permutation with repetitions, and without repeated elements $\left( n, k\right)$-permutation without repetitions.

The Newton’s binomial and polynomial expansion are considered.