Problem 5

In a circus,

n

clowns dress and paint themselves using a selection of 12 distinct colours. Each clown must use at least five different colours. The ringmaster requires that no two clowns have exactly the same set of colours and that no more than 20 clowns use any one particular colour. Find the largest possible

n

Step 3 of 6: Derive the upper bound

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5n\le12\cdot20=240\Longrightarrow n\le48

Detailed analysis

Combining the two incidence bounds gives $5n\le12\cdot20=240$ , so every valid arrangement has $n\le48$ .