Counting

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Counting

• By Patrick, Andrew, Jared Jonathan

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Counting

• Counting is the study of the enumeration of
  discrete, finite sets (counting objects)

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• Ex. If there are 5 symbols with which we are
  given, to create a 4 symbol license plate, how
  many plates can we make? (Symbols can be
  repeated)
• (5)(5)(5)(5) 625

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Multiplication Principle for Counting

• If an activity can be performed as a sequence of
  k independent steps, and step i can be
  accomplished in ni ways, than the entire
  activity can be performed in
• (n1)(n2)(n3)(nk) ways.
• When can this be used?

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• Ex. If we must create a license plate in which
  two of the symbols can be of a given set of 6
  letters or two of the symbols can be of a given
  set of 5 numbers, how many unique plates can we
  make?
• (5)(5) (6)(6) 61

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Addition Principle for Counting

• If an activity can be divided into k disjoint
  subsets of activities (separate activities), and
  the ith of these sets contains ni elements, than
  total number is n1 n2 n3. nk.

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Permutation

• A permutation of n distinct objects x1, x2,
  x3xn is any ordering of n objects.
• P(n,r) n!/(n-r)! where one chooses r items
  from n items (order matters).
• Where does it come from?

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Example

• If there are 16 baseball players who need to be
  put into a 3 player batting order, how many
  different batting orders are possible?
• P(16,3) 16!/(16-3)! 16!/13! 3,360.
• There are 3,360 unique batting orders which are
  possible.

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Combinations

• Combinations are Permutations where the order of
  the objects being counted doesnt matter.
• For example 1, 2, and 3 can be arranged in 6
  different ways if order matters but only 1 way if
  order does matter

Matters Doesnt Matter
1,2,3 1,3,2 2,1,3 2,3,1 3,2,1 3,1,2
1,2,3
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Combinations Contd

• The table from the previous page shows that for a
  set of (1,2,3) if you want to order the numbers
  with a Permutation you will get 6 outputs, but
  with a Combination, you only receive 1.
• This shows that there are 6 times as many
  Permutations than combinations for this set.
  This works for any P(n,r) and C(n,r).
• Since there are 3 digits, you divide the number
  of permutations by another 3! which is equal to 6.

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Combinations Equation

• To reach this conclusion we reduce the formula by
  how many ways the objects could be in order which
  accounts for the r! on the bottom.
• P(n,r) C(n,r)r!

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Combinations with repetition

• Combinations with repetition are also referred to
  as Multichoose problems.
• The equation is
• How does it work?

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Multichoose

• Lets say you have 5 different flavors of ice
  cream, and you can have 3 scoops. Your 3 scoops
  can be any flavor, with repetition, how many
  different combinations of ice cream scoops can
  you have?
• We will say the flavors are banana, chocolate,
  lemon, strawberry and vanilla.

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Multichoose Contd

• Think about the ice cream being in containers,
  you could say "skip the first, then 3 scoops,
  then skip the next 3 containers" and you will end
  up with 3 scoops of chocolate!
• So there are r (n-1) positions, and we want to
  choose r of them to have circles.

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Example

• If we refer back to the previous problem about
  batting orders, we will take the same number of
  players, 16, but this time 3 players will be
  picked to be team captains. How many ways can
  they be chosen in groups of 3?
• C(16,3) 16!/3!(16-3)!
• 16!/3!(13)!
• 161514/321
• 560
• In this case, there are only 560 ways that
  players can be chosen, because order does not
  matter.

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Identity 130

• Identity For 0 k n
• k(nk)n(n-1k-1)
• Question How many ways can we create a size k
  committee of students from a class of n students,
  where one of the committee members is designated
  as the chair?

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Identity 130 continued

• There are (nk) ways to choose the committee, then
  k ways to select a chair. Therefore, there are
  k(nk) possible outcomes.
• First, select the chair from the class of n
  students. Then, from the remaining n-1 students,
  pick the remaining k-1 members. This can be done
  n(n-1k-1) ways.

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Identity 131

• We have a question we could ask
• How many ways can we make a committee (of any
  size) from a class of students of n students,
  where one of them is designated as the chair.
• Starting this we will ask how many committees we
  could make.
• This would be the same as (nk).
• We must then factor in the people who may be the
  chair.
• This is k.

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Identity 131 continued

• Another way to solve the same thing
• We see that there are n students to be the chair
  of committees, so we select the chair of each
  subcommittee out of n students.
• There are now n-1 students remaining.
• It follows that there are 2n-1 ways of filling
  the remaining committees.
• We now have n2n-1 ways of choosing different
  ways of choosing committees.
• If we look at the identity we were given, this
  helps show that the answer we got is correct.

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Thank you to our mentor

• Greg Warrington!

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References Slide!

• Proofs That Really Count Benjamin/Quinn
• Introduction to Discrete Mathematics,
  Burgmeier/Kost

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Homework Problems

• We have a baseball team made up of 6 players.
• A. How many different 3 person batting orders can
  we have? (order matters, Permutation)
• B. How many different 2 person groups of team
  captains can we have? (order doesnt matter,
  Combination)