1. The problem statement, all variables and given/known data Question: How many simple substitution ciphers are there where no point is fixed (ie: no letter is mapped to itself)? EDIT: Incase the termonoligy is different, a simple substitution cipher is a mapping where the plaintext in english is encoded so that every letter is mapped to a different letter. IE: a maps to Z b maps to S c maps to K etcetcetc thus abc plaintext is mapped to ZSK. 3. The attempt at a solution So, I did a few examples to see if I could get some insight into the problem: 1 Letter = 0 mappings 2 Letters = 1 mapping 3 Letters = 2 mappings 4 Letters = 9 mappings 5 Letters = 44 mappings I stopped here While I found the reason for the complexity of determining this number I didn't come across any way to do it in general for 26 letters (for English). I know it has to be above (n-1)! and below n! for n letters but other then that I'm unsure. I'm thinking I can just brute force it using algebraic notation, for example in the case of 4 letters I counted the mappings: 1 - 4cycle and with 4 letters there are 6 distinct combinations 2 - 2cycles and with 4 letters there are 3 distinct combinations But, with 26 letters the number of different cycle combinations becomes cumbersome. Any guidance would be appreciated.