Code and ciphers: Julius Caesar, the Enigma and the internet

(2) In how many ways can N be represented as the sum of (any number of)
positive integers where the order of the integers is relevant? This
number is denoted by c(N) and is called the number of combinations
of N. For example:
4:�4�3�1�1�3�2�2�2�1�1�1�2�1�1�1�2�1�1�1�1
so that c(4)�8. In fact it can be proved that
c(N)�2 (N�1) ,
for a proof of which see [10.2].
(3) In how many ways can N be represented as the sum of k (a fixed number
of) positive integers when the order of the integers is relevant?
Since there are k integers and each of these must be greater than or
equal to 1 the total number is given by the coefficient of X N in the
expansion of
X k (1�X) �k
or, what is the same thing, the coefficient of X (N�k) in the expansion of
(1�X) �k
and this is
(N � 1)!
(k � 1)!(N � k)
a formula which is relevant to Problem 4.2 ( where N�9 and k�3,
giving the value 28).
M19 Maximum multiple of the kick which can occur
when differencing Hagelin key
Consider a wheel of length w with a kick of k. When we difference the
pattern at any distance, d, which is not a multiple of w, there are four possibilities,
shown in Table A.1.
Table A.1
Pin N Pin (N�d) Difference of key values
Inactive Inactive 0
Inactive Active �k
Active Inactive �k
Active Active 0
Mathematical aspects 207