Find the sums of the following series:
a) \({\frac {1} {1 \times 2}} + {\frac {1} {2 \times 3}} + {\frac {1} {3 \times 4}} + {\frac {1} {4 \times 5}} + \dots\);
b) \({\frac {1} {1 \times 2 \times 3}} + {\frac {1} {2 \times 3 \times 4}} + {\frac {1} {3 \times 4 \times 5}} + {\frac {1} {4 \times 5 \times 6}} + \dots\);
c) \({\frac {0!} {r!}} + {\frac {1!} {(r-1)!}} + {\frac {2!} {(r-2) !}} + {\frac {3!} {(r-3)!}} + \dots\) for \(r \geq 2\).
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) \(4: 0\)?
Consider a chess board of size \(n \times n\). It is required to move a rook from the bottom left corner to the upper right corner. You can move only up and to the right, without going into the cells of the main diagonal and the one below it. (The rook is on the main diagonal only initially and in the final moment in time.) How many possible routes does the rook have?
Tickets cost 50 cents, and \(2n\) buyers stand in line at a cash register. Half of them have one dollar, the rest – 50 cents. The cashier starts selling tickets without having any money. How many different orders of people can there be in the queue, such that the cashier can always give change?
Prove that the Catalan numbers satisfy the recurrence relationship \(C_n = C_0C_{n-1} + C_1C_{n-2} + \dots + C_{n-1}C_0\). The definition of the Catalan numbers \(C_n\) is given in the handbook.
Determine all prime numbers \(p\) and \(q\) such that \(p^2 - 2q^2 = 1\) holds.
Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of \(d\). Prove that \(d >30,000\).