On a chessboard (an \(8 \times 8\) grid), we place eight identical rooks. A rook can move any number of squares in a straight line horizontally (along a row) or vertically (along a column). In chess, a piece can take another piece if it can move to the other piece’s square in a single move.
In how many ways can we arrange the eight rooks so that no rook can take any other?
How many five-digit numbers are there which are written 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.
A set is a collection of elements where each element appears only once. The elements are not ordered, and there is no rule connecting them. Even a set with no elements (an empty set) counts as a set. The collections \(\{a,b,c,d\}\) and \(\{3,2,45,1,0,\pi\}\) are both examples of sets.
Let \(C\) be a set with \(n\) elements. How many different sets can be formed using the elements of \(C\)?
There are six letters in the alphabet of the Gloops. 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 Gloops?
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 \(11n \leq k \leq m/2\). Prove that then there are at least \(mn\) pairs of people in the country who will not be able to talk without interpreters.
A group of \(15\) elves decided to pay a visit to their relatives in a distant village. They have a horse carriage that fits only \(5\) elves. In how many ways can they assemble the ambassador team, if at least one person in the team needs to be able to operate the carriage, and only \(5\) elves in the group can do that?
There are \(5\) pirates and they want to share \(8\) identical gold coins. In how many ways can they do it if each pirate has to get at least one coin?
Is \(100\times 99 \times ...\times 2\) divisible by \(2^{100}\)?
It is known that \(a + b + c = 5\) and \(ab + bc + ac = 5\). What are the possible values of \(a^2 + b^2 + c^2\)?