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) 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?
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?