In Carl\textquotesingle s pencil case there are nine pencils. At least one of the pencils is blue. In any group of four pencils, at least two have the same colour. In any group of five pencils, at most three have the same colour. How many pencils are blue? 
Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations
The information that in any group of four pencils, at least two have the same colour, tells us that there at most three different coloured pencils in Carl\textquotesingle s pencil case. The information that in any group of five pencils, at most three have the same colour, tells us that there are at most three pencils of any single colour in the pencil case. Hence there are three pencils of each of the three different colours and so Carl\textquotesingle s pencil case contains three blue pencils. 