Problems

A coin is tossed three times. How many different sequences of heads and tails can you get?

Each cell of a \(2 \times 2\) square can be painted either black or white. How many different patterns can be obtained?

In a football team (made up of 11 people), a captain and his deputy need to be chosen. How many ways can this be done?

If a salary is first increased by 20%, and then reduced by 20%, will the salary paid increase or decrease as a result?

Arrange in a row the numbers from 1 to 100 so that any two neighbouring ones differ by at least 50.

The numbers from 1 to 9999 are written out in a row. How can I remove 100 digits from this row so that the remaining number is a) maximal b) minimal?

A group of 20 tourists go on a trip. The oldest member of the group is 35, the youngest is 20. Is it true that there are members of the group that are the same age?

Two grandmasters in turn put rooks on a chessboard (one turn – one rook) so that they cannot capture each other. The person who cannot put a rook on the chessboard loses. Who will win with the game – the first or second grandmaster?

In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?