Combinations

Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route?

My coat has three buttons. How many ways can you find to do up all the buttons?

Weekly Problem 21 - 2011
How many ways can you paint this wall with four different colours?

How many 10-digit numbers containing only 1s, 2s and 3s can you write?

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

What is the smallest number of ferry trips that Neda needs to take to visit all four islands and return to the mainland?

Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

From the atomic masses recorded in a mass spectrometry analysis can you deduce the possible form of these compounds?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Can all but one square of an 8 by 8 Chessboard be covered by Trominoes?