This post contains a spoiler for problem C from the Facebook HackerCup finals. If you still want to upsolve this, you might not want to read further.

I really liked this problem but in the end was a bit disappointed that in the end we could not compute the actual probability. What we compute has no interpretation and only proves to the problem setters that we are able to compute this "answer" efficiently. But what is then the usecase of having an algorithm that quickly computes a nonsensical number? If it has no use in practice, then it demotivates me to learn how to solve these kind of large combinatorics questions.

Thinking a bit more about this... we could make a custom BigInteger class using 1000 bit integers and then we could compute the actual probability, right? Is this the reason that it is still useful to be able to solve a question like this efficiently?