Lesson 2

In the summary list below, \(k=2\) objects are selected from \(n=3\) options \(\Sigma=\{{\tt a},{\tt b},{\tt c}\}\). In combinatorics, the urn scheme is also used to describe all four types of objects: there is an urn containing \(n=3\) balls \(\Sigma=\{{\tt a},{\tt b},{\tt c}\}\), and we sequentially draw \(k=2\) balls from it; we may or may not return the ball to the urn, and the order of the drawn balls may or may not matter.

• Arrangements with repetition.  Order matters, elements can repeat. Also known as: a word (or an element of) \(\Sigma^{k}\).

  from itertools import product
  
  for p in product('abc', repeat=2):
      print(*p, sep='', end=' ')

  
  

  aa ab ac ba bb bc ca cb cc
  



• Arrangements without repetition.  Order matters, elements cannot repeat. Also known as: \(k\)-permutation.

  from itertools import permutations
  
  for p in permutations('abc', 2):
      print(*p, sep='', end=' ')

  
  

  ab ac ba bc ca cb
  



• Combinations with repetition.  Order does not matter, elements can repeat. Also known as: \(k\)-multiset.

  from itertools import combinations_with_replacement
  
  for p in combinations_with_replacement('abc', 2):
      print(*p, sep='', end=' ')

  
  

  aa ab ac bb bc cc
  



• Combinations without repetition.  Order does not matter, elements cannot repeat. Also known as: \(k\)-set.

  from itertools import combinations
  
  for p in combinations('abc', 2):
      print(*p, sep='', end=' ')

  
  

  ab ac bc