There is an elastic band and glass beads: four identical red ones, two identical blue ones and two identical green ones. It is necessary to string all eight beads on the elastic band in order to get a bracelet. How many different bracelets can be made so that beads of the same colour are not next to each other? (Assume that there is no buckle, and the knot on the elastic is invisible).
What has a greater value: \(300!\) or \(100^{300}\)?
A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
How many are there six-digit numbers that are divisible by \(5\)?
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) \(4: 0\)?
Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.