Is it possible to fill a \(5 \times 5\) board with \(1 \times 2\) dominoes?
In each cell of a \(25 \times 25\) square table, one of the numbers 1, 2, 3, ..., 25 is written. In cells, that are symmetric relative to the main diagonal, equal numbers are written. There are no two equal numbers in any row and in any column. Prove that the numbers on the main diagonal are pairwise distinct.
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?
How many ways can Rob fill in one card in the “Sport Forecast” lottery? (In this lottery, you need to predict the outcomes of thirteen sports matches. The result of each match is the victory of one of the teams or a draw, and the scores do not play a role).
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?
There are five books on a shelf. In how many ways can the books be arranged in a stack. (Stacks may consist of any number of books)?
\(N\) young men and \(N\) young ladies gathered on the dance floor. How many ways can they split into pairs in order to participate in the next dance?
There are three groups of stones: in the first – 10, in the second – 15, in the third – 20. During one turn, you are allowed to split any pile into two smaller ones; the one who cannot make a move loses.
Numbers from 1 to 20 are written in a row. Players take turns placing pluses and minuses between these numbers. After all of the gaps are filled, the result is calculated. If it is even, then the first player wins, if it is odd, then the second player wins. Who won?