Fall in Love

It's really a hard time at DUCS, because we're gonna miss our loved seniors. There are $n$ seniors at DUCS and for each $i$ $(1 \leq i \leq n)$, the $i^{th}$ senior visits DUCS $A_{i}$ times every $30$ days, but it's not at all concrete so nothing can be said firmly in this regard. The probability of finding all the seniors at DUCS is known as the _Fall in Love_ probability $P$. Surprisingly, aft