The school cafeteria offers three varieties of pancakes and five different toppings. How many different pancakes with toppings can Emmanuel order? He has to have exactly one topping on each pancake.
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?
On a chessboard (an \(8 \times 8\) grid), we place eight identical rooks. A rook can move any number of squares in a straight line horizontally (along a row) or vertically (along a column). In chess, a piece can take another piece if it can move to the other piece’s square in a single move.
In how many ways can we arrange the eight rooks so that no rook can take any other?
How many five-digit numbers are there which are written 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.
A set is a collection of elements where each element appears only once. The elements are not ordered, and there is no rule connecting them. Even a set with no elements (an empty set) counts as a set. The collections \(\{a,b,c,d\}\) and \(\{3,2,45,1,0,\pi\}\) are both examples of sets.
Let \(C\) be a set with \(n\) elements. How many different sets can be formed using the elements of \(C\)?
There are six letters in the alphabet of the Gloops. 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 Gloops?
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, 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 \(n\).