Multiplication%20Principle%20and%20Addition%20Principle

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Multiplication Principle and Addition Principle

• Multiplication Principle Suppose a task is
  accomplished by n steps and each step requires a
  choice from a number of available choices. Let
  these numbers be A1, A2, . . . An
• Then, the total number of ways to
  accomplish this task is
• A1 x A2 x A3 x . .
  . . An.

• Addition Principle Suppose a task is
• accomplished by choosing an object from the
  union of disjoint sets with cardinalities B1, B2,
  B3, . .. Bn. Then, the total number of ways to
  accomplish this task is
• B1 B2 B3 . . . . . Bn.

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Set Theoretic Descriptionsof the two principles

• The cardinality of the Cartesian product of n
  sets A1, A2, . . An is given by
• n (A1 x A2 x . .. An) n(A1). n(A2) . .
  .n(An).
• The cardinality of the disjoint union of n sets
• B1, B2, . . . . Bn is given by
• n (B1 U B2 U . . . Bn) n(B1) n(B2) . .
  . n(Bn)

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Examples

• On a table, there is a pile of 10 apples and
  another pile of 15 pears.
• 1. How many choices do you have if you are
  instructed to pick an apple and a pear?
• 2. How many choices do you have if you are
  instructed to pick an apple or a pear?

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Mistakes you must avoid

• Suppose there are 4 doctors and 5 chess players
  in a room with 7 people. If you are instructed to
  pick a doctor or a chess player in this room (and
  slap the face) how many choices do you have?
• Is your answer 45 9 ? Then, review your
  knowledge of addition principle.

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Read the problems carefully.

• In a room, there are 7 people. Everyone in
  that room is either a chess player or a doctor. 4
  of them are doctors and 5 of them are chess
  players. In how many ways can one accomplish each
  of the following tasks?
• 1. Task is to get into the room, pick a chess
  player and play a chess with. Then, pick a doctor
  and ask for an opinion about your asthma symptom.
• 2. Task is to pick a team of two people,
  consisting of one chess player and one doctor.

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• Addition Principle is frequently applied in a
  form of subtraction principle.
• n(A \ B) n(A) n(B)
• Example In a basket of 30 Easter eggs, there are
  7 green eggs. Your task is to pick an egg that is
  not green. How many choices do you have? The
  answer 30-7 23. 23 choices.
• Another Example (See the Venn diagram of chess
  players and doctors.) We will approach the
  second question in the previous
• slide in the following way. Initially, we
  write a name of a chess player and (next to it)
  write a name of a doctor. Consider this writing
  as a tentative list of 2-team members. There are
  5x420 possible ways of writing a pair of names
  in this way. But, exactly 2 of them are a pair of
  names of a same person (which is not permitted).
  So, the answer to the second question ( of
  forming a team of chess_player-doctor pair) is
  20-218.

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Multiplication Principle is frequently applied in
the form of Division Principle.

• Division Principle
• If AB x C (cartesion product of B and
  C), then
• n(B) n(A) / n(C).
• EXAMPLE 1 Gregor Samsa goes to walk every
  morning wearing a hat and carrying a cane. He has
  2 hats, which are identical except that one is
  grey and the other is brown. He has 3 canes. He
  is color blind. In how many ways, can he choose
  a hat and a cane to go out for a walk based on
  his discerning ability?
• Answer If he were able to tell brown from
  grey, there would be
• 2 x 3 6 ways. But, now
  that all the hats are identical to him
• from his view, there are 6 /
  2 3 ways.

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• Division Principle re-stated
• Suppose a task A consists of accomplishing a
  task B followed by a task C. Then, the number of
  ways to accomplish task B is given by the number
  of ways to accomplish the task A divided by the
  number of ways to accomplish task C.
• EXAMPLE 2. How many different words can be
  formed by rearranging the letters in the word
  ELEMENT?
• Answer 7! / 3! ( Explain. . . . .)

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Permutation

• Informally, a permutation on a set of n elements
  is an ordering of these elements.
• Formally, a permutation on a set of n elements is
  a one-to-one correspondence between the set and
  itself.
• Number of permutations on a set of n elements
• Generalized Permutation Let A be a set of n
  elements and rlt n. Any one-to-one function from
  the set
• 1,2,3, . .. . r into the set A is
  called a generalized permutation of the k
  elements on the set A.
• Number of generalized permutation of r elements
  on the set of n elements is denoted by nPr.
• nPr x (n-r)! n!
• by multiplication
  principle.

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Combination

• Let rltn. An r-combination on the set of n
  elements is a subset with cardinality r.
• nCr denotes the number of all the r-combinations
  on the set of n elements.
• nPr and nCr are related by the following equation
• nPr nCr x r! (illustrated below)

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• Picking a generalized permutation,
• a one-to-one function from 1,2, . . r into
• The set A of n elements is equivalent to
• STEP1 Picking a subset of A that has r
  elements.
• STEP 2 Picking a permutation on these r
  elements.
• Therefore,
• n P r n C r x r !

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Summary

• n! number of permutations on n objects.
• nPr number of r-permutations on n objects
• nCr number of r-sets that can be formed from n
  objects
• nPr / r!

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Partition

• Let A be a set of n elements. If, for some
  subsets A1, A2, . . . , Ak that are pairwise
  disjoint,
• A A1 U A2 U . . . U
  Ak
• then we say A1, A2, . . . Ak is a
  partition of A.

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Ordered Partition

• Let a1a2 . . . ak n and a1lt a2lt . . ak
• ( lt here means less than or equal to)
• The sequence of subsets
• A1, A2, . . . . Ak with cardinalities
• a1, a2, . . . . . ak
• is called an ordered partition if these sets
  are mutually disjoint and their union is the set
  A.

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Number of ordered partitions

• How many ordered partitions of a given type does
  a set of n element have?
• Experiment Consider a set of 7 elements. How
  many ordered partitions of type
• (2,2,3) are there?
  The construction of a partition of a given
• type can be
  considered as a multi-step task. The number of
• these steps is 1 less
  than the number of subsets that form the
• partition. In this
  particular experiment, 2 steps.
• STEP 1 Pick a subset of 2
  elements (from 7 elements)
• STEP 2 Pick a subset of 2
  elements (from the remaining 5 elements)
• Now, we employ the multiplication
  principle
• 7 C 2 x 5 C 2

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Counting Ordered Partitions of a Given Type

• Let a1a2 . . . ak n. The
  number of type (a1, a2, . . . . ak) partitions
  of a set of n elements is given by

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Application (of partition counting)

• How many different words can be formed by
  rearranging the letters in the word
• MISSISSIPI ?
• Discussion Compare this problem with the
  following problem How many ordered partitions
  of the type (1,1,4,4) are there for a set of 10
  elements?.
• How are these two problems related?

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• Given a 10 element set, imagine these elements
  are doors along a hall way of a hotel. Imagine
  your duty is to write
• letter M on 1 door
• letter P on 1 door
• letter S on 4 doors
• letter I on 4 doors
• This job is equivalent to forming a word by an
  arrangement of letters in MISSISSIPI.
• This job is also equivalent to forming a type
  (1,1,4,4) ordered partition of the 10 element
  set.