Too many valentines

Discrete Mathematics Level 5

Jack is an immortal handsome young man who lives in a galaxy far far away. This galaxy, however celebrates valentine's day as well.

Jack, being the heartthrob he is, gets a new valentine every year but decides to keep the ones he had his previous years as well. However, Jack is a bit of a geek and doesn't want just anybody to be his valentine. So he devises a test to make sure that he would only celebrate valentine's day that year if each and every one of his dates can solve a problem posed by him.

Jack numbers each of his valentine's from \(1\) to \(n\) [during year n] and also takes \(n\) pieces of paper and writes down these numbers. He then takes \(n\) envelopes and randomly puts each paper into an envelope and numbers the envelopes randomly as well. He would then give his valentine's each a chance to go find the envelope with their number in it. However, each of them are only allowed to open \(\frac{n}{2}\) envelopes and have to place the numbers back in the same envelope after they're done. They must then close the envelope. They are not allowed to communicate until everyone's turn is over.

Jack will only celebrate valentine's day with all of his \(n\) dates if every single one of them can find their number. If atleast one of his valentine's is unable to find their envelope, he will cancel all his dates.

Given that all of Jack's valentine's are smart and use the best possible approach, what is the probability that Jack will not cancel all his dates towards the end of his immortal lifetime? (as \(n\) approaches \( \infty \) )


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