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?
How many six-digit numbers exist, the numbers of which are either all odd or all even?
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?
The Russian Chess Championship is made up of one round. How many games are played if 18 chess players participate?
Prove that the product of any three consecutive natural numbers is divisible by 6.
Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).
Between the nine planets of the solar system, a cosmic messaging system is introduced. Rockets fly along the following routes: Earth – Mercury, Pluto – Venus, Earth – Pluto, Pluto – Mercury, Mercury – Venus, Uranus – Neptune, Neptune – Saturn, Saturn – Jupiter, Jupiter – Mars and Mars – Uranus. Is it possible to get from Earth to Mars?
The board has the form of a cross, which is obtained if corner boxes of a square board of \(4 \times 4\) are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?