On a board there are written 10 units and 10 deuces. During a game, one is allowed to erase any two numbers and, if they are the same, write a deuce, and if they are different then they can write a one. If the last digit left on the board is a unit, then the first player won, if it is a deuce then the second player wins.
Two players in turn put coins on a round table, in such a way that they do not overlap. The player who can not make a move loses.
A pawn stands on one of the squares of an endless in both directions chequered strip of paper. It can be shifted by \(m\) squares to the right or by \(n\) squares to the left. For which \(m\) and \(n\) can it move to the next cell to the right?
a) Two students need to be chosen to participate in a mathematical Olympiad from a class of 30 students. In how many ways can this be done?
b) In how many ways can a team of three students in the same class be chosen?
How many ways can Susan choose 4 colours from 7 different ones?
On the plane, 10 points are marked so that no three of them lie on the same line. How many triangles are there with vertices at these points?
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)?
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?