Basic Combinatorics

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Basic Combinatorics
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The Fundamental Counting Principle

• How many ways can you pick one letter of the
  alphabet?
• 26
• How many ways can you pick one letter of the
  alphabet and one single digit?
• 2610 260

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The Fundamental Counting Principle
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The Fundamental Counting Principle

• Let E1 and E2 be two events. The first event E1
  can occur in m1 different ways. After E1 has
  occurred, E2 can occur in m2 different ways. The
  number of ways that the two events can occur is
  m1 x m2
• In our example the first event was to pick a
  letter of the alphabet and there were 26 ways of
  doing that. The second event was to pick a
  single digit and we had 10 ways to do that. The
  total number of ways that we could pick both was
  26 x 10 or 260.

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The Fundamental Counting Principle

• Lets take a look at a common lottery problem.
• Say that you need to pick 5 numbers to win and
  the range is from 0 to 42. (Numbers cannot be
  repeated.) How many different possibilities are
  there?

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The Fundamental Counting Principle

• How many ways can we pick the first number (this
  is event E1)?
• How many ways can we pick the second number
  (event E2)?
• And so on

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The Fundamental Counting Principle

• Lets take a look at another lottery problem.
• Say that you need to pick 5 numbers to win and
  the range is from 0 to 42. This time, the
  numbers are thrown back in the hopper and can be
  selected again. How many different possibilities
  are there?

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The Fundamental Counting Principle

• How many ways can we pick the first number (this
  is event E1)?
• How many ways can we pick the second number
  (event E2)?
• And so on

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The Fundamental Counting Principle

• The first lottery example was without
  replacement and the second was with
  replacement.

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Permutations

• A permutation of n different elements is an
  ordering of the elements such that one element is
  first, one is second, one is third and so on.
• This is an application of the Fundamental
  Counting Principle.

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Permutation Example

• How many different permutations are possible for
  the letters A, B, C, D, E, and F?
• First position Six possible letters
• Second position Five possible letters
• Third position Four possible letters
• Fourth position Three possible letters
• Fifth position Two possible letters
• Sixth position One possible letter

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Permutation Formula

• Permutations of n Elements Taken r at a Time

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Permutation Example

• There are eight horses in a race. In how many
  different ways can these horses come in first,
  second, and third?

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Distinguishable Permutations

• Suppose a set of n objects has n1 of one kind of
  object, n2 of a second kind, n3 of a third kind,
  and so on with n n1 n2 n3 nk. Then the
  number of distinguishable permutation of the n
  objects is

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Distinguishable Permutations Example

• How many distinguishable ways can the letters in
  BANANA be written?

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Combinations

• I always think of combinations as committees.
  (because order is not important).
• In the horse racing example, order was important
  ABC and BAC are two different possible outcomes.
• In committee selection, order is not important.
  Two groups with the same members in different
  order are the same. ABC and BAC are the same
  committee.

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Combination Formula

• Notice that this is the same formula we used for
  binomial expansion! Cool. But why?

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Combination Example

• A standard poker hand consists of five cards
  dealt from a deck of 52. How many different
  poker hands are possible?

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Counting Problems

• Students sometimes find them difficult because
  they forget to pay attention to whether or not
  the order is important.
• Stop. Take a deep breath and figure out what is
  going on.