A person has 10 friends and within a few days invites some of them to visit so that his guests never repeat (on some of the days he may not invite anyone). How many days can he do this for?
How many ways can you cut a necklace consisting of 30 different beads into 8 pieces (you can cut only between beads)?
30 people vote on five proposals. In how many ways can the votes be distributed if everyone votes only for one proposal and only the number of votes cast for each proposal is taken into account?
How many necklaces can be made from five identical red beads and two identical blue beads?
a) The sports club has 30 members, of which four people are required to participate in the 1,000 metre race. How many ways can this be done?
b) How many ways can I build a team of four people to participate in the relay race 100 m + 200 m + 300 m + 400 m?
How many ways can you build a closed line whose vertices are the vertices of a regular hexagon (the line can be self-intersecting)?
Find the number of rectangles made up of the cells of a board with \(m\) horizontals and \(n\) verticals that contain a cell with the coordinates \((p, q)\).
Each of the edges of a complete graph consisting of 6 vertices is coloured in one of two colours. Prove that there are three vertices, such that all the edges connecting them are the same colour.
A class has more than 32, but less than 40 people. Every boy is friends with three girls, and every girl is friends with five boys. How many people are there in the class?
A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?