Problems

Age
Difficulty
Found: 2016

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?

How many ways can you choose four people for four different positions, if there are nine candidates for these positions?

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) \(\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n}\).

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 \(0 = x = y < z\).

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.

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 \(ABC\)?