Prove that there are infinitely many natural numbers \(\{1,2,3,4,...\}\).
Prove that there are infinitely many prime numbers \(\{2,3,5,7,11,13...\}\).
Is it possible to colour the cells of a \(3\times 3\) board into red and yellow such that there are the same number of red cells and yellow cells?
Sometimes one can guess certain multiples of a number just by looking at it, the idea of this sheet is to learn to recognise quickly using tricks when a natural number is divisible by another number.
Today we will study the method of finding the amount of combinations, or consecutive actions, or ways to select items from a bag which is called the Product rule. The main idea of this combinatorial is the following: if you are asked to perform an action that can be done in, say \(5\) ways and another action afterwards that can be done in \(4\) ways, then the total number of possibilities to perform two consecutive actions would be equal to \(5\times 4\). The reason for this is the opportunity to choose \(4\) possible second actions for each of the \(5\) choices of the first action already made before.
A coin is tossed six times. How many different sequences of heads and tails can you get?
Each cell of a \(3 \times 3\) square can be painted either black, or white, or grey. How many different ways are there to colour in this table?
Consider a set of numbers \(\{1,2,3,4,...n\}\) for natural \(n\). Find the number of permutations of this set, namely the number of possible sequences \((a_1,a_2,...a_n)\) where \(a_i\) are different numbers from \(1\) to \(n\).
Eleven people were waiting in line in the rain, each holding an umbrella. They stood closely together, so that the umbrellas of the neighbouring people were touching (see fig.)
The rain stopped and all people closed their umbrellas. They now stood keeping a distance of \(50\) cm between neighbours. By how many times has the queue length decreased? People can be considered points, and umbrellas are circles with a radius of \(50\) cm.
Cut a square into five triangles in such a way that the area of one of these triangles is equal to the sum of the area of other four triangles.