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Let $X = \{1, 2, 3, \ldots , 25\}$. If a student selects a function randomly from the set of all functions from $X$ onto $X$, then what is the probability that the selected function maps prime numbers to prime numbers?

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Start by figuring out how many prime numbers there are in your set. Also consider the the set {1,2,3,4,5} and try to to work it out for this small set, then generalize. –  Alex R. Apr 27 '12 at 20:59
    
@ Sam ok I'll try. –  Abhishek Pant Apr 27 '12 at 21:01
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up vote 3 down vote accepted

Hint: a function from $X$ onto $X$ (where $X$ is a finite set) is a permutation of $X$. If $A$ and $B$ are the sets of primes and non-primes in $X$, a permutation of $X$ that maps $A$ onto $A$ (and therefore $B$ onto $B$) corresponds to an arbitrary permutation of $A$ and an arbitrary permutation of $B$. Now, how many are there?

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