Each of 5 rotors may be chosen for the 3 rotor slots. Each rotor creates a permutation of the code. They would have been smart not to have any of the 5 rotor permutations be commlutable with any other, so for choices for first 3 slots one gets (all in agreement with this web site):
5*4*3=5!/2!=60.
Then obvious settings of the 26 position rotors:
26*26*26=26^3=17576.
Now plug board: Apparently from pictures they didn't use jumpers blocks, so the plugs must indeed have internal switches so that a position without a plug shorts through from output to input. Each cable enacts a swap, but all the 10 cables are identical. Likewise swapping ends of the cables produces an identical results.
26*25/2 * 24*23/2 * 22*21/2 * ... * 10*9/2 * 8*7/2 / 10! =
(26!/6!)/(10!*2^10) = 158.962555217826 million million million ish, call it "159" like in the movie! (PS: Not one scene of The Imitation Game came from the book, not one thing actually happened. Still I think that the decoder wheels bore some vague relationship to the "bombs" construction and is useful for thought. Successive columns would be successive input/rotor positions, and rows correspond to rotors all tree searched. Needs more for searching the plugboard settings than was shown in the movie. At least I'm making some progress in understanding this now.)
Now the 26*26 additional ring settings are not "accounted for" in any other calculation. They do truly establish another multiplier of settins positions. HOWEVER the decoding will on average go 13 characters before a "notch" kicks the 2nd wheel over, so one could see if there was an appearance of a recognizable word most likely. Thus these don't have to be searched for as intently as a fractional word or phrase can lead to decoding the notch or ring positions later as the first breakpoint into unintelligibility. Thus leaving this out is just a convention.
Also note that there are actually more possible plug board settings because the code sheets can indicate that a cable is not used, so one actually has 10 cables, 9 cables, 8 cables, etc. They probably would not go too far down.
Each of 5 rotors may be chosen for the 3 rotor slots. Each rotor creates a permutation of the code. They would have been smart not to have any of the 5 rotor permutations be commlutable with any other, so for choices for first 3 slots one gets (all in agreement with this web site):
5*4*3=5!/2!=60.
Then obvious settings of the 26 position rotors:
26*26*26=26^3=17576.
Now plug board: Apparently from pictures they didn't use jumpers blocks, so the plugs must indeed have internal switches so that a position without a plug shorts through from output to input. Each cable enacts a swap, but all the 10 cables are identical. Likewise swapping ends of the cables produces an identical results.
26*25/2 * 24*23/2 * 22*21/2 * ... * 10*9/2 * 8*7/2 / 10! =
(26!/6!)/(10!*2^10) = 158.962555217826 million million million ish, call it "159" like in the movie! (PS: Not one scene of The Imitation Game came from the book, not one thing actually happened. Still I think that the decoder wheels bore some vague relationship to the "bombs" construction and is useful for thought. Successive columns would be successive input/rotor positions, and rows correspond to rotors all tree searched. Needs more for searching the plugboard settings than was shown in the movie. At least I'm making some progress in understanding this now.)
Now the 26*26 additional ring settings are not "accounted for" in any other calculation. They do truly establish another multiplier of settins positions. HOWEVER the decoding will on average go 13 characters before a "notch" kicks the 2nd wheel over, so one could see if there was an appearance of a recognizable word most likely. Thus these don't have to be searched for as intently as a fractional word or phrase can lead to decoding the notch or ring positions later as the first breakpoint into unintelligibility. Thus leaving this out is just a convention.
Also note that there are actually more possible plug board settings because the code sheets can indicate that a cable is not used, so one actually has 10 cables, 9 cables, 8 cables, etc. They probably would not go too far down.