Counting Number of Possible Solutions

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Counting Number of Possible Solutions Simple
Combinatorics
Combination - an un-ordered collection of
distinct elements, usually of a prescribed size
and taken from a given set. Given such a set S, a
combination of elements of S is just a subset of
S, where as always for (sub)sets the order of the
elements is not taken into account (two lists
with the same elements in different orders are
considered to be the same combination). Also, as
always for (sub)sets, no elements can be repeated
more than once in a combination this is often
referred to as a "collection without repetition".

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Counting Number of Possible Solutions Simple
Combinatorics

• Permutations - a sequence containing each element
  from a finite set once, and only once. The
  concept of sequence is distinct from that of a
  set, in that the elements of a sequence appear in
  some order the sequence has a first element
  (unless it is empty), a second element (unless
  its length is less than 2), and so on. In
  contrast, the elements in a set have no order
  1, 2, 3 and 3, 2, 1 are different ways to
  denote the same set.
• However, there is also a traditional more
  general meaning of the term "permutation" used in
  combinatorics. In this more general sense,
  permutations are those sequences in which, as
  before, each element occurs at most once, but not
  all elements of the given set need to be used.

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Counting Number of Possible Solutions Simple
Combinatorics

• Combination the number of combinations that can
  be made when choosing k items from a set of n
  items is denoted as
• Example how many different hands of 5 cards can
  be dealt from a pack of 52 cards?

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Counting Number of Possible Solutions Simple
Combinatorics

• Permutation the number of permutations that can
  be made from a set of n items is n!
• Or in general, the number of permutations of size
  r that can be formed from a set of n items is
• Example how many sequences can you form from
  the numbers 1,2,3?
• Answer 3! 6
• 1-2-3 1-3-2 2-1-3 2-3-1 3-1-2 3-2-1

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Counting Number of Possible Solutions Simple
Combinatorics
Traveling salesman problem For the term project
consisting of 48 cities, how many possible
solutions exist? Answer 48! 1.24 x 1061
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Counting Number of Possible Solutions Simple
Combinatorics
Knapsack problem For the 20 item knapsack
homework problem, how many possible solutions
exist? Answer 20 190 1140
184756 ... 1140 190 20 1