The probability for a random permutation of n = 4,

First proof:
There are precisely 10 permutationen of {1,2,3,4}
which have no cycle of length 3 or 4, namely:
(1)(2)(3)(4)  
(12)(3)(4)  (13)(2)(3)  (14)(2)(3) 
(23)(1)(4)  (24)(1)(3)  (34)(1)(2) 
(12)(34)  (13)(24)  (14)(23) 
Note also here: If every prisoner chooses randomly 2 drawers, then
the probability that all prisoners are pardoned, is
(0,5)^{4} = 0,0625 ≈ 6 %