Calculate the following sums:
a)
b)
c)
In the expansion of
Show that any natural number
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?
Here is a fragment of the table, which is called the Leibniz triangle. Its properties are “analogous in the sense of the opposite” to the properties of Pascal’s triangle. The numbers on the boundary of the triangle are the inverses of consecutive natural numbers. Each number is equal to the sum of two numbers below it. Find the formula that connects the numbers from Pascal’s and Leibniz triangles.
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?
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?
Each side in the triangle
How many integers are there from 1 to 1,000,000, which are neither full squares, nor full cubes, nor numbers to the fourth power?
Consider a chess board of size