SlideServe
Télécharger Obtenir Pro
Menu
Accueil Pro
Préférences
ApparenceLight Aide et commentaires
© SlideServe · v2026

Review of Basic Counting

ulric avatar
ulric
20 diapositives · 329 Views · Sep 26, 2012

This comprehensive guide explores fundamental principles of counting, focusing on permutations and combinations. Starting with basic counting methods using cards, it delves into sample problems involving award distributions and critical scenarios like bomb defusion. The text highlights the significance of order in arrangements and introduces special cases in permutations. Readers will learn about the total number of arrangements, the formula for permutations, and multiple principles that govern combinations, providing a solid foundation for understanding these critical mathematical concepts.

#basic english #stocktaking #judicial review #account #countdown #survey #recension #rodeo
Review of Basic Counting - slide 1
01 / 20Review of Basic Counting
1 / 17

Télécharger la présentation

Review of Basic Counting
Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation

Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement. Téléchargez la présentation en cliquant sur ce lien. Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.

F I N

Plus de contenu associé

Transcription de la présentation

    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) 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)

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