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I'm not sure I've seen this puzzle before, and I really like it:

We have n keys and n boxes. Each key fits only one box. We shuffle the keys and put one in each box. Then we randomly break open 1≤k≤n boxes. What is the probability that we can unlock all the other boxes?

@christianp
If I understand this correctly, then it is zero unless $k>=n/2$. Since otherwise you dont have enough keys, right?

Spoilers 

@christianp For k=1, it would be 1/n, since there's an (n-a)/(n-a+1) chance that the aᵗʰ box you open doesn't have the key to the box you broke open, and the product of those fractions gives 1/n.

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