Two players take turns to put rooks on a chessboard so that the rooks cannot capture each other. The player who cannot make a move loses.
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?
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 necklaces can be made from five identical red beads and two identical blue beads?
How many ways can you build a closed line whose vertices are the vertices of a regular hexagon (the line can be self-intersecting)?
How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,
a) if each number can occur only once?
b) if each number can occur several times?
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?