Review of Basic Counting

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Review of Basic Counting

• Suppose there are 10 cards in a box,
• the number 0, 1, 2, , 9 was written
• on each card. How many ways to pick
• 2 cards from the box such that the
• sum of 2 numbers (on the cards) is
• odd if you
• 1) pick 2 cards at the same time.
• 2) pick 2 cards one by one.

5x525 ways
252550 ways
2
???????????????????????????

• ???????? ?????? 10 ?? ???????????
  ????????????????????? 0 1 2 3 9
  ????????????????????? 2 ?????????????????????????
  ?????????? ???
• 1) ???????????? 2 ?? (?????? ??)
• 2) ?????????? (???????)

3
Review of Basic Counting

• 100! ???????????????????

4
agenda

• Permutations
• Combinations
• Some derivation of Permutations and Combinations
• Eliminating Duplicates
• r-Combinations with Repetitions

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Permutations

• Sample problems
• Five athletes (Amazon, Bobby, Corn, Dick and
  Ebay) compete in an Olympic event. Gold, silver
  and bronze medals are awarded in how many ways
  can the awards be made?

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Permutation Example

• A terrorist has planted an armed nuclear bomb in
  your city, and it is your job to disable it by
  cutting wires to the trigger device. There are
  10 wires to the device. If you cut exactly the
  right three wires, in exactly the right order,
  you will disable the bomb, otherwise it will
  explode! If the wires all look the same, what
  are your chances of survival?

1098 720, so there is a 1 in 720 chance that
youll survive!
????????????
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Permutation (cont.)

• Order matters !!!
• The case that Amazon wins gold and Ebay wins
  silver is different from the case Ebay wins gold
  and Amazon wins silver.
• If the order is of significance, the
  multiplication rules are often used when several
  choices are made from one and the same set of
  objects.

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Permutations-Definition

• In general, if r objects are selected from a set
  of n objects, any particular arrangement of these
  r objects(say, in a list) is called a
  permutation.
• the total number of permutations of r objects
  selected from a set of n objects is nPr or P(n,
  r)
• In other words, a permutation is an ordered
  arrangement of objects.

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Permutations-formal

• A permutation of a set S of objects is a sequence
  containing each object once.
• An ordered arrangement of r distinct elements of
  S is called an r-permutation.

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Permutations More examples

• Examples
• How many permutations of 3 of the first 5
  positive integers are there?
• How many permutations of the characters in
  COMPUTER are there? How many of these end in a
  vowel?

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Ex1 Let Sa, b, c, d, find all permutations of
3 elements selected from set S

• 24 permutations

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??????????????????

• ??????????????????????????
• ????????????????????????
• ???????????????????????
• ???????????????????????????

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• By multiple principle, the total number of
  permutations of r objects selected from a set of
  n objects is
• n(n-1)(n-2)(n-r1)
• Using factorial
• P(n, r)

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Permutations -- Special Cases

• Using CAT
• P(n,0)
• Theres only one ordered arrangement of zero
  objects, the empty set.
• P(n,1)
• There are n ordered arrangements of one object.
• P(n,n)
• There are n! ordered arrangements of n distinct
  objects (multiplication principle)

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• Note We just only focus on finding the numbers
  of arrange r distinct things from n distinct
  things linearly.

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Think!!!

• ??????? ??? S ????????????????? ???????????????
  f S ? S ??????????????????? S (permutation on S)
  ??? f ???????????????????????????? S ????????? S
• ??? Sym(S) ???????????????????????????????? S
  ??? Sn ????????????????????????????????????? S
  ??????????? n ??? ??? f,g ???????????????????? S
  ???????? f ??? g ???????????????? fg
  ????????????????? gof

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Exercise

• ?????????(???????????????????)???
  ??????????????????????? SECOND ??????????
  (permutations of SECOND?)
• ??????????????????? 3 ???? ???????????? 1, 3, 5
  ??? 7 ??????????? ???????????????????????????
• ??????????????????????? 8 ???? ????????? 4 ????
  ??????????????????????? 8 ?????????? ???
• 4 ???????????????????????
• ??????????????????????????
• ??????????????????????????????
• ??????????????????????????????????????????
• ??????????????????????????????????????????????????
  ?

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???????

• ????? 5 ?? ????????????? ????????????
  ?????????????????????????????????? ??????????
• ????????? 10 ????????????????? ?????????????? 4
  ?? ?????????? ??????????
• ????????? 2 ??? ????????????????????? 1 ????
  ?????????????? (inclusion-exclusion principle)

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??????? 10 ??????????? ??????? 4
?????????????????????????? 1 ???? ??????????????
(inclusion-exclusion principle)

• ??????????????????????????????????????????? ????
  ?????????? 7 ??????????? 3 ?? ????????????
• ??? Ai ??????????????????????? i ?????? ???????
  ????? ???????

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More Exercises

• ??????????? ??? ?????? 3 ?? ????????? 2 ?? (???
  7 ??) ??????????????????? ?????????? ???
  ????????????????? ??????????????????????????
  ?????????????????????????????????
• ??????????????????? 7 ?? ?????????????????????????
  ???????? ??? ??? A ?????? B (?????????? 7 ?????)
  ?????????????
• ??????????????????? 3 ???? ???? 3 ???? ?????? 3
  ???? ????????????????? ???????????????????????????
  ????????????????? ?????????????
  ?????????????????????????????????
  ??????????????????????????????????????????????????
  ????????