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silk road castings

Source: SRMC 2013 P4

September 2, 2018
combinatorics

Problem Statement

In the film there is nn roles. For each ii (1≤i≤n1 \le i \le n), the role of number ii can play aia_i a person, and one person can play only one role. Every day a casting is held, in which participate people for nn roles, and from each role only one person. Let pp be a prime number such that p≥a1,…,an,np \ge a_1, \ldots, a_n, n. Prove that you can have pkp^k castings such that if we take any kk people who are tried in different roles, they together participated in some casting (kk is a natural number not exceeding nn ).
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