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?
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?
There are 9 cities in the country Number with the names 1, 2, 3, 4, 5, 6, 7, 8, 9. The traveller discovered that two cities are connected by an airline if and only if a two-digit number made up of the digit-names of these cities, is divisible by 3. Is it possible to get from city 1 to city 9?
In a state there are 100 cities, and from each of them there are 4 roads. How many roads are there in the state?
In the country Seven there are 15 cities, each of which is connected by roads with no less than seven other cities. Prove that from every city you can get to any other city (possibly passing through other cities).
There is a group of islands connected by bridges so that from each island one can get to any of the other islands. The tourist has bypassed all the islands, walking on each bridge exactly once. He visited the island of Three-isle three times. How many bridges are there on Three-isle if the tourist
a) did not start on it and did not finish on it?
b) started on it, but did not finish on it?
c) started on it and finished on it?