MathLabs

Problem 5

In a circus, nn 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 nn.
Step 1 of 6: Count users of each colour
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Ei={clowns using colour i},∣Ei∣≤20E_i=\{\text{clowns using colour }i\},\qquad |E_i|\le20
Detailed analysis

For each colour ii, let EiE_i be the set of clowns using it. The rule says ∣Ei∣≤20|E_i|\le20 for every ii, so ∑i=112∣Ei∣≤12⋅20\sum_{i=1}^{12}|E_i|\le12\cdot20.