There are \(n\) points on the plane. How many lines are there with endpoints at these points?
On a plane \(n\) randomly placed lines are given. What is the number of triangles formed by them?
On two parallel lines \(a\) and \(b\), the points \(A_1, A_2, \dots , A_m\) and \(B_1, B_2, \dots , B_n\) are chosen, respectively, and all of the segments of the form \(A_iB_j\), where \(1 \leq i \leq m\), \(1 \leq j \leq n\). How many intersection points will there be if it is known that no three of these segments intersect at one point?
How many six-digit numbers exist, for which each succeeding number is smaller than the previous one?
How many four-digit numbers can be made using the numbers 1, 2, 3, 4 and 5, if:
a) no digit is repeated more than once;
b) the repetition of digits is allowed;
c) the numbers should be odd and there should not be any repetition of digits?
In a box, there are 10 white and 15 black balls. Four balls are removed from the box. What is the probability that all of the removed balls will be white?
Write at random a two-digit number. What is the probability that the sum of the digits of this number is 5?
There are three boxes, in each of which there are balls numbered from 0 to 9. One ball is taken from each box. What is the probability that
a) three ones were taken out;
b) three equal numbers were taken out?
A player in the card game Preferans has 4 trumps, and the other 4 are in the hands of his two opponents. What is the probability that the trump cards are distributed a) \(2: 2\); b) \(3: 1\); c) \(4: 0\)?
Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.
Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).