Deal two packs of shuffled cards simultaneously. What is the probability that no pair of identical cards will be exposed simultaneously?
Fix the first pack, and consider all possible rearrangements of the
second pack. For each i in the range 
let 
denote the set of all arrangements of the second pack which
happen to have the property that the card in position i matches
the card in position i of the first pack. Obviously
for every 
Moreover, if 
then 
and so on. Let 
so
the probability
of at least one match is 
We calculate this
using the inclusion-exclusion principle.
Thus the probability of no coincidences is (to an excellent approximation)
Here we have use the fact that
and put 