Problems

Annie and Hanna are preparing some Christmas baubles. They want to paint each bauble all in one colour. They have \(7\) different colours of paint and \(26\) baubles to paint. In how many ways can they do this? Two ways are considered the same if the numbers of baubles of each colour are the same. Each bauble has to be painted but not all the colours need to be used.

An \(8 \times 8\) square is divided into \(1 \times 1\) cells. It is covered with right-angled isosceles triangles (two triangles cover one cell). There are 64 black and 64 white triangles. We consider "regular" coverings - such that every two triangles having a common side are of a different colour. How many "regular" covers are there?

You are given a pentagon \(ABCDE\) such that \(AB = BC = CD = DE\), and \(\angle B = \angle D = 90^\circe\). Show how the plane can be tiled with pentagons equal to the given one.

A family is going on a big holiday, visiting Austria, Bulgaria, Cyprus, Denmark and Estonia. They want to go to Estonia before Bulgaria. How many ways can they visit the five countries, subject to this constraint?

How many subsets of \(\{1,2,...,n\}\) (that is, the integers from \(1\) to \(n\)) have an even product? For the purposes of this question, take the product of the numbers in the empty set to be \(1\).

In the following grid, how many different ways are there of getting from the bottom left triangle to the bottom right triangle? You must only go from between triangles that share an edge and you can visit each triangle at most once. (You don’t have to visit all of the triangles.)

Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).

How many subsets are there of \(\{1,2,...,n\}\) (the integers from \(1\) to \(n\) inclusive) containing no consecutive digits? That is, we do count \(\{1,3,6,8\}\) but do not count \(\{1,3,6,7\}\).