I have currently been studying discrete mathematics and combinatorics where I came across the introduction to generalized permutations in the textbook (Introductory Discrete Mathematics by V.K. Balakrishnan if that helps) in the chapter on combinatorics. I understand the derivation of the initial formula, of which is P(n; n1, n2, ... nk) = (n!)/((n1)!(n2)!...(nk)!) where n is the number of elements in set X and n (subscript) i (i = 1, 2, 3, ... ,k) is the number of elements of X in group i. The book then quickly gives another formula of which is equivalent of the form P(n, r)/((n1)!(n2)!...(nk)!) and furthermore derives from this the following three statements:
1) P(n; n1, n2, ..., n(k-1)) = P(n; n1, n2, ... , n(k-1), m) where m = (n-(n1+n2+...+n(k-1))
2)P(n; r) = P(n; n-r) = P(n; r, n-r)
3) (r!)P(n; r) = P(n, r)
What these three statements inherently mean and the derivation of them is something I would greatly appreciated.