7. Combinatorial algorithms and decomposition of  numbers [1.3]

*** Permutations [1.3.1]

А. No repetition  n = 3 { 1 2 3 } - ordered set of n elements.
123  132  213  231  312  321
Total number: n!
[permute.cpp]

Б. With repetition n = 4 { 1 1 2 3 } - ordered multiset of n elements
1123  1132  1213  1231  1312  1321  2113  2131  2311  3112  3121  3211
Total number: n!/(s₁!s₂!...s_k!), where s_i! is the number of i-th different element in the multiset,
For the example: 4!/(2!1!1!) = 12

*** k-permutations of n (Variations) [1.3.2] - ordered k-element list of n elements

A. No repetition n = 3 { 1 2 3 },  k = 2
12  13  21  23  31  32
Total number n!/(n - k)! 
For the example: 3!/1 = 6

B. With repetitions n = 3 { 1 2 3 },  k = 2
11  12  13  21  22  23  31  32  33
Total number n^k
For the example: 3² = 9
[variate.cpp]

*** Combinations [1.3.3] - k-element subset of n elements (unordered)

A. No repetition  n = 5 { 1 2 3 4 5 },  k = 2
12  13  14  15  23  24  25  34  35  45
Total number n!/((n - k)!k!)
For the example: 5!/(3!2!) = 10
[comb.cpp]

B. With repetition n = 5 { 1 2 3 4 5 },  k = 2
11  12  13  14  15  22  23  24  25  33  34  35  44  45  55
Total number (n + k - 1)! / ((n - 1)! k!)
For the example:   6!/(4!2!) = 15