Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.
On a plane, six points are given so that no three of them lie on the same line. Each pair of points is connected by a blue or red segment.
Prove that among these points three such points can be chosen so that all sides of the triangle formed by them will be of the same colour.
There are 17 carriages in a passenger train. How many ways can you arrange 17 conductors around the carriages if one conductor has to be in each carriage?
The number of permutations of a set of \(n\) elements is denoted by \(P_n\).
Prove the equality \(P_n = n!\).
How many ways can you choose four people for four different positions, if there are nine candidates for these positions?
Out of two mathematicians and ten economists, it is necessary to form a committee made up of eight people. In how many ways can a committee be formed if it has to include at least one mathematician?
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?
Prove the validity of the following formula of Newton’s binom \[(x+y)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}y + \dots + \binom{n}{n}y^n.\]