SILK-ROAD 2013 Problem 4

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