How many six-digit numbers are there whose digits all have the same parity? That is, either all six digits are even, or all six digits are odd.
Donald’s sister Maggie goes to a nursery. One day the teacher at the nursery asked Maggie and the other children to stand a circle. When Maggie came home she told Donald that it was very funny that in the circle every child held hands with either two girls or two boys. Given that there were five boys standing in the circle, how many girls were standing in the circle?
In how many ways can eight rooks be arranged on the chessboard in such a way that none of them can take any other. The color of the rooks does not matter, it’s everyone against everyone.
How many five-digit numbers are there which are written in the same from left to right and from right to left? For example the numbers \(54345\) and \(12321\) satisfy the condition, but the numbers \(23423\) and \(56789\) do not.
Definition A set is a collection of elements, containing only one copy of each element. The elements are not ordered, nor they are governed by any rule. We consider an empty set as a set too.
There is a set \(C\) consisting of \(n\) elements. How many sets can be constructed using the elements of \(C\)?
There are six letters in the alphabet of the Bim-Bam tribe. A word is any sequence of six letters that has at least two identical letters. How many words are there in the language of the Bim-Bam tribe?
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 \(n\).
A rectangular parallelepiped of the size \(m\times n\times k\) is divided into unit cubes. How many rectangular parallelepipeds are formed in total (including the original one)?