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Review of Basic Counting

???????????????????????????. ???????? ?????? 10 ?? ??????????? ????????????????????? 0 1 2 3 ? 9 ????????????????????? 2 ????????????????????????? ?????????? ???1) ???????????? 2 ?? (?????? ??)2) ?????????? (???????). 100! ???????????????????. Review of Basic Counting. agenda. PermutationsComb

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Review of Basic Counting

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    1. 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.

    2. ??????????????????????????? ???????? ?????? 10 ?? ??????????? ????????????????????? 0 1 2 3 … 9 ????????????????????? 2 ????????????????????????? ?????????? ??? 1) ???????????? 2 ?? (?????? ??) 2) ?????????? (???????)

    3. 100! ???????????????????

    4. agenda Permutations Combinations Some derivation of Permutations and Combinations Eliminating Duplicates r-Combinations with Repetitions

    5. 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?

    6. 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?

    7. 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.

    8. 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.

    9. 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.

    10. 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?

    11. Ex1 Let S={a, b, c, d}, find all permutations of 3 elements selected from set S 24 permutations

    12. ?????????????????? ?????????????????????????? ???????????????????????? ??????????????????????? ???????????????????????????

    13. 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-r+1) Using factorial P(n, r) =

    14. Permutations -- Special Cases Using CAT P(n,0) There’s 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)

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

    16. 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

    17. Exercise ?????????(???????????????????)??? ??????????????????????? SECOND ?????????? (permutations of SECOND?) ??????????????????? 3 ???? ???????????? 1, 3, 5 ??? 7 ??????????? ??????????????????????????? ??????????????????????? 8 ???? ????????? 4 ???? ??????????????????????? 8 ?????????? ??? 4 ??????????????????????? ?????????????????????????? ?????????????????????????????? ?????????????????????????????????????????? ???????????????????????????????????????????????????

    18. ??????? ????? 5 ?? ????????????? ???????????? ?????????????????????????????????? ?????????? ????????? 10 ????????????????? ?????????????? 4 ?? ?????????? ?????????? ????????? 2 ??? ????????????????????? 1 ???? ?????????????? (inclusion-exclusion principle)

    19. ??????? 10 ??????????? ??????? 4 ??????????????????????????????????? ?????????????? (inclusion-exclusion principle) ??????????????????????????????????????????? ???? ?????????? 7 ??????????? 3 ?? ???????????? ??? Ai ??????????????????????? i ?????? ??????? ????? ???????

    20. More Exercises ??????????? ??? ?????? 3 ?? ????????? 2 ?? (??? 7 ??) ??????????????????? ?????????? ??? ??????????????????????? ?????????????????????????? ????????????????????????????????? ??????????????????? 7 ?? ????????????????????????????????? ??? ??? A ?????? B (?????????? 7 ?????) ????????????? ??????????????????? 3 ???? ???? 3 ???? ?????? 3 ???? ????????????????? ???????????????????????????????????????????? ????????????? ????????????????????????????????? ??????????????????????????????????????????????????????????

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