4 prisoners, each one being allowed to open 2 drawers:

The probability for a random permutation of n = 4,
    that all cycles have length at most 2, is ≈ 42 %.


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)


Second proof: The calculation as in case n=100 yields:

 

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 %