Problems

In how many ways can eight rooks be arranged on the chessboard in such a way that none of them can take any other. The color of the rooks does not matter, it’s everyone against everyone.

How many five-digit numbers are there which are written in the same from left to right and from right to left? For example the numbers $54345$ and $12321$ satisfy the condition, but the numbers $23423$ and $56789$ do not.

Definition A set is a collection of elements, containing only one copy of each element. The elements are not ordered, nor they are governed by any rule. We consider an empty set as a set too.
There is a set $C$ consisting of $n$ elements. How many sets can be constructed using the elements of $C$?

There are six letters in the alphabet of the Bim-Bam tribe. A word is any sequence of six letters that has at least two identical letters. How many words are there in the language of the Bim-Bam tribe?

A coin is tossed six times. How many different sequences of heads and tails can you get?

Each cell of a $3\times 3$ square can be painted either black, or white, or grey. How many different ways are there to colour in this table?

Consider a set of numbers $\{1,2,3,4,...n\}$ for natural $n$. Find the number of permutations of this set, namely the number of possible sequences $(a_1,a_2,...a_n)$ where $a_i$ are different numbers from $1$ to $n$.

A rectangular parallelepiped of the size $m\times n\times k$ is divided into unit cubes. How many rectangular parallelepipeds are formed in total (including the original one)?

In the Land of Linguists live $m$ people, who have opportunity to speak $n$ languages. Each person knows exactly three languages, and the sets of known languages may be different for different people. It is known that $k$ is the maximum number of people, any two of whom can talk without interpreters. It turned out that $11n\le k\le m/2$. Prove that then there are at least $mn$ pairs of people in the country who will not be able to talk without interpreters.

Eleven people were waiting in line in the rain, each holding an umbrella. They stood closely together, so that the umbrellas of the neighbouring people were touching (see the figure)

The rain stopped and everyone closed their umbrellas. They then shuffled closer, keeping a distance of $50$ cm between neighbours. What is the ratio of the old queue length to the new queue length? People can be considered points, and umbrellas are circles with a radius of $50$ cm.