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?
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?
On the plane, 10 points are marked so that no three of them lie on the same line. How many triangles are there with vertices at these points?
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?