Calculate the following sums:
a) \(\binom{5}{0} + 2\binom{5}{1} + 2^2\binom{5}{2} + \dots +2^5\binom{5}{5}\);
b) \(\binom{n}{0} - \binom{n}{1} + \dots + (-1)^n\binom{n}{n}\);
c) \(\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n}\).
In the expansion of \((x + y)^n\), using the Newton binomial formula, the second term was 240, the third – 720, and the fourth – 1080. Find \(x\), \(y\) and \(n\).
Show that any natural number \(n\) can be uniquely represented in the form \(n = \binom{x}{1} + \binom{y}{2} + \binom{z}{3}\) where \(x, y, z\) are integers such that \(0 \leq x < y < z\), or \(0 = x = y < z\).
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 \(ABC\) is divided into 8 equal segments. How many different triangles exist with the vertices at the points of division (the points \(A\), \(B\), \(C\) cannot be the vertices of triangles) in which neither side is parallel to either side of the triangle \(ABC\)?
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 \(n \times n\). 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?