Ce diaporama a bien été signalé.
Nous utilisons votre profil LinkedIn et vos données d’activité pour vous proposer des publicités personnalisées et pertinentes. Vous pouvez changer vos préférences de publicités à tout moment.
Basic Quantitative Techniques ABS-Bangalore Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010
<ul><li>Dr. R. Venkatamuni Reddy Associate Professor </li></ul><ul><li>Contact: 09632326277, 080-30938181 [email_address] ...
Permutations  and  Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
Permutations <ul><li>Permutations refers to the different ways in which a number of a number of objects can be arranged in...
<ul><li>“ The word permutation thus refers to the arrangements which can be made by taking some or all of a number of thin...
<ul><li>Example 1: There are six boxes and three balls. In how many ways can these three balls be discretely put into thes...
<ul><li>Example 3: How many arrangements are possible of the letters of the words ‘JAIPUR’, ‘BANGALORE’ and ‘ALLIANCE’  </...
<ul><li>Formulae 2:  Finding the number of permutations of ‘n’ things taken ‘r’ at a time, given that each of the elements...
<ul><li>Solution: n=11, p=2 ( as letter B is occurring twice in the given word) , and q=2 ( as letter I is occurring twice...
Permutations Basic Quantitative Techniques - RVMReddy - ABS
Permutation formula proof <ul><li>There are  n  ways to choose the first element </li></ul><ul><ul><li>n -1 ways to choose...
<ul><li>Combinations refers to the number of arrangements which can be made from a group of things irrespective of the ord...
<ul><li>Formulae 1:  The number of  r -combinations of a set with  n  elements, where  n  is non-negative and 0≤ r ≤ n  is...
<ul><li>Example 1: in how many ways can four persons be chosen out of seven? </li></ul><ul><li>n=7, r=4 </li></ul><ul><li>...
<ul><li>Formulae 2: The number of ways in which x+y+z things can be divided into three groups contain x, y, and z things r...
Combinations Basic Quantitative Techniques - RVMReddy - ABS
Combinations Basic Quantitative Techniques - RVMReddy - ABS
Combinations <ul><li>How many different poker hands are there (5 cards)? </li></ul><ul><li>How many different (initial) bl...
Combination formula proof <ul><li>Let  C (52,5) be the number of ways to generate unordered poker hands </li></ul><ul><li>...
Combination formula proof <ul><li>Let  C ( n , r ) be the number of ways to generate unordered combinations </li></ul><ul>...
Combination formula proof <ul><li>Note that the textbook explains it slightly differently, but it is same proof </li></ul>...
<ul><li>Let  n  and  r  be non-negative integers with  r  ≤  n .  Then  C ( n , r ) =  C ( n , n-r ) </li></ul><ul><li>Pro...
Binomial Coefficients <ul><li>The expression  x  + y  is   a  binomial expression  as it is the sum of two terms.  </li></...
Binomial Coefficients Basic Quantitative Techniques - RVMReddy - ABS
Binomial Coefficients Basic Quantitative Techniques - RVMReddy - ABS
Binomial Coefficients <ul><li>Pascal’s Triangle  </li></ul><ul><ul><li>The number  C(n ,  r)  can be obtained   by constru...
Binomial Coefficients July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
Basic Quantitative Techniques - RVMReddy - ABS
Binomial Coefficients <ul><li>The technique known as  divide and conquer  can be used   to compute  C(n ,  r ).  </li></ul...
Thank You Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010
Prochain SlideShare
Chargement dans…5
×

Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages

15 884 vues

Publié le

  • DOWNLOAD FULL BOOKS, INTO AVAILABLE FORMAT ......................................................................................................................... ......................................................................................................................... 1.DOWNLOAD FULL. PDF EBOOK here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. EPUB Ebook here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. doc Ebook here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. PDF EBOOK here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. EPUB Ebook here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. doc Ebook here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... .............. Browse by Genre Available eBooks ......................................................................................................................... Art, Biography, Business, Chick Lit, Children's, Christian, Classics, Comics, Contemporary, Cookbooks, Crime, Ebooks, Fantasy, Fiction, Graphic Novels, Historical Fiction, History, Horror, Humor And Comedy, Manga, Memoir, Music, Mystery, Non Fiction, Paranormal, Philosophy, Poetry, Psychology, Religion, Romance, Science, Science Fiction, Self Help, Suspense, Spirituality, Sports, Thriller, Travel, Young Adult,
       Répondre 
    Voulez-vous vraiment ?  Oui  Non
    Votre message apparaîtra ici
  • DOWNLOAD FULL BOOKS, INTO AVAILABLE FORMAT ......................................................................................................................... ......................................................................................................................... 1.DOWNLOAD FULL. PDF EBOOK here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. EPUB Ebook here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. doc Ebook here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. PDF EBOOK here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. EPUB Ebook here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... 1.DOWNLOAD FULL. doc Ebook here { https://tinyurl.com/y6a5rkg5 } ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... .............. Browse by Genre Available eBooks ......................................................................................................................... Art, Biography, Business, Chick Lit, Children's, Christian, Classics, Comics, Contemporary, Cookbooks, Crime, Ebooks, Fantasy, Fiction, Graphic Novels, Historical Fiction, History, Horror, Humor And Comedy, Manga, Memoir, Music, Mystery, Non Fiction, Paranormal, Philosophy, Poetry, Psychology, Religion, Romance, Science, Science Fiction, Self Help, Suspense, Spirituality, Sports, Thriller, Travel, Young Adult,
       Répondre 
    Voulez-vous vraiment ?  Oui  Non
    Votre message apparaîtra ici
  • DOWNLOAD FULL. BOOKS INTO AVAILABLE FORMAT ......................................................................................................................... ......................................................................................................................... 1.DOWNLOAD FULL. PDF EBOOK here { https://tinyurl.com/y8nn3gmc } ......................................................................................................................... 1.DOWNLOAD FULL. EPUB Ebook here { https://tinyurl.com/y8nn3gmc } ......................................................................................................................... 1.DOWNLOAD FULL. doc Ebook here { https://tinyurl.com/y8nn3gmc } ......................................................................................................................... 1.DOWNLOAD FULL. PDF EBOOK here { https://tinyurl.com/y8nn3gmc } ......................................................................................................................... 1.DOWNLOAD FULL. EPUB Ebook here { https://tinyurl.com/y8nn3gmc } ......................................................................................................................... 1.DOWNLOAD FULL. doc Ebook here { https://tinyurl.com/y8nn3gmc } ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... .............. Browse by Genre Available eBooks ......................................................................................................................... Art, Biography, Business, Chick Lit, Children's, Christian, Classics, Comics, Contemporary, Cookbooks, Crime, Ebooks, Fantasy, Fiction, Graphic Novels, Historical Fiction, History, Horror, Humor And Comedy, Manga, Memoir, Music, Mystery, Non Fiction, Paranormal, Philosophy, Poetry, Psychology, Religion, Romance, Science, Science Fiction, Self Help, Suspense, Spirituality, Sports, Thriller, Travel, Young Adult,
       Répondre 
    Voulez-vous vraiment ?  Oui  Non
    Votre message apparaîtra ici
  • its nice
       Répondre 
    Voulez-vous vraiment ?  Oui  Non
    Votre message apparaîtra ici

Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages

  1. 1. Basic Quantitative Techniques ABS-Bangalore Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010
  2. 2. <ul><li>Dr. R. Venkatamuni Reddy Associate Professor </li></ul><ul><li>Contact: 09632326277, 080-30938181 [email_address] [email_address] </li></ul>Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010
  3. 3. Permutations and Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  4. 4. Permutations <ul><li>Permutations refers to the different ways in which a number of a number of objects can be arranged in a different order </li></ul><ul><li>Example: Suppose there are two things x and y, they can be arranged in to two different ways i.e,. xy and yx . These two arrangements is called permutation </li></ul><ul><li>Similarly x, y and z </li></ul><ul><li>xyz, xzy, yxz, yzx, zxy, zyx is 6 arrange permutation </li></ul><ul><li>(if we want to have two things only from x,y,z then xy,xz,yz,yx,zx,yz only in this case) </li></ul>July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  5. 5. <ul><li>“ The word permutation thus refers to the arrangements which can be made by taking some or all of a number of things” </li></ul><ul><li>Formulae 1: Finding the number of permutations of ‘n’ dissimilar things taken ‘r’ at a time </li></ul><ul><li>n=number of different things given, r=number of different things taken at a time out of different things given </li></ul>Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  6. 6. <ul><li>Example 1: There are six boxes and three balls. In how many ways can these three balls be discretely put into these six boxes. </li></ul><ul><li>Solution: </li></ul><ul><li>Example 2: How many four-letter words can be made using the letters of the word ‘BANGALORE’ and ‘ALLIANCE’ </li></ul><ul><li>Solution: n=9, r=4 and n=8, r=4 </li></ul>Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  7. 7. <ul><li>Example 3: How many arrangements are possible of the letters of the words ‘JAIPUR’, ‘BANGALORE’ and ‘ALLIANCE’ </li></ul><ul><li>Hint: n=6, r=6 and n=9, r=9 and n=8,r=8 </li></ul>Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  8. 8. <ul><li>Formulae 2: Finding the number of permutations of ‘n’ things taken ‘r’ at a time, given that each of the elements can be repeated once, twice….up to ‘r’ times </li></ul><ul><li>Or </li></ul><ul><li>‘ n’ things taken all at a time of which ‘p’ are alike, ‘q’ others are alike and ‘r’ others alike </li></ul><ul><li>Example 1: How many permutations are possible of the letters of the word PROBABILITY when taken all at a time? </li></ul>Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  9. 9. <ul><li>Solution: n=11, p=2 ( as letter B is occurring twice in the given word) , and q=2 ( as letter I is occurring twice in the given word) </li></ul><ul><li>And all other letters in the given word are different. The required number of permutations is (r is not valid in this) </li></ul><ul><li>Example 2: You are given a word “MANAGEMENT” and asked to compute the number of permutations that you can form taking all the letters from this word? </li></ul>Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  10. 10. Permutations Basic Quantitative Techniques - RVMReddy - ABS
  11. 11. Permutation formula proof <ul><li>There are n ways to choose the first element </li></ul><ul><ul><li>n -1 ways to choose the second </li></ul></ul><ul><ul><li>n -2 ways to choose the third </li></ul></ul><ul><ul><li>… </li></ul></ul><ul><ul><li>n - r +1 ways to choose the r th element </li></ul></ul><ul><li>By the product rule, that gives us: </li></ul><ul><li>P ( n , r ) = n ( n -1)( n -2)…( n - r +1) </li></ul>July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  12. 12. <ul><li>Combinations refers to the number of arrangements which can be made from a group of things irrespective of the order </li></ul><ul><li>Combinations differ from permutations in that one combination such as xyz may be stated in the form of several permutations just by rearranging the orders as : xyz, xzy, yxz, yzx, zxy, zyx </li></ul><ul><li>Note: All of these are one combination but they are six permutations </li></ul><ul><li>IMP Note: The number of permutations is always greater than the number of combinations in any given situation since a combination of n different things can be generate n factorial permutations </li></ul>Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  13. 13. <ul><li>Formulae 1: The number of r -combinations of a set with n elements, where n is non-negative and 0≤ r ≤ n is: </li></ul><ul><li>n= number of different things given </li></ul><ul><li>r= number of different things taken at a time out of different things given </li></ul>Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  14. 14. <ul><li>Example 1: in how many ways can four persons be chosen out of seven? </li></ul><ul><li>n=7, r=4 </li></ul><ul><li>Example 2: Find the number of combinations of 50 things taking 46 at a time. ANS: 230300 </li></ul>Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  15. 15. <ul><li>Formulae 2: The number of ways in which x+y+z things can be divided into three groups contain x, y, and z things respectively is </li></ul><ul><li>Example: In how many ways can 10 books be put to three shelves which can contain 2, 3 and 5 books respectively? </li></ul>Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  16. 16. Combinations Basic Quantitative Techniques - RVMReddy - ABS
  17. 17. Combinations Basic Quantitative Techniques - RVMReddy - ABS
  18. 18. Combinations <ul><li>How many different poker hands are there (5 cards)? </li></ul><ul><li>How many different (initial) blackjack hands are there? </li></ul>July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  19. 19. Combination formula proof <ul><li>Let C (52,5) be the number of ways to generate unordered poker hands </li></ul><ul><li>The number of ordered poker hands is P (52,5) = 311,875,200 </li></ul><ul><li>The number of ways to order a single poker hand is P (5,5) = 5! = 120 </li></ul><ul><li>The total number of unordered poker hands is the total number of ordered hands divided by the number of ways to order each hand </li></ul><ul><li>Thus, C (52,5) = P (52,5)/ P (5,5) </li></ul>July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  20. 20. Combination formula proof <ul><li>Let C ( n , r ) be the number of ways to generate unordered combinations </li></ul><ul><li>The number of ordered combinations (i.e. r -permutations) is P ( n , r ) </li></ul><ul><li>The number of ways to order a single one of those r -permutations P ( r,r ) </li></ul><ul><li>The total number of unordered combinations is the total number of ordered combinations (i.e. r -permutations) divided by the number of ways to order each combination </li></ul><ul><li>Thus, C ( n,r ) = P ( n,r )/ P ( r,r ) </li></ul>July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  21. 21. Combination formula proof <ul><li>Note that the textbook explains it slightly differently, but it is same proof </li></ul>July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  22. 22. <ul><li>Let n and r be non-negative integers with r ≤ n . Then C ( n , r ) = C ( n , n-r ) </li></ul><ul><li>Proof: </li></ul>Combination formula proof July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  23. 23. Binomial Coefficients <ul><li>The expression x + y is a binomial expression as it is the sum of two terms. </li></ul><ul><li>The expression (x + y) n is called a binomial expression of order n . </li></ul>Basic Quantitative Techniques - RVMReddy - ABS
  24. 24. Binomial Coefficients Basic Quantitative Techniques - RVMReddy - ABS
  25. 25. Binomial Coefficients Basic Quantitative Techniques - RVMReddy - ABS
  26. 26. Binomial Coefficients <ul><li>Pascal’s Triangle </li></ul><ul><ul><li>The number C(n , r) can be obtained by constructing a triangular array. </li></ul></ul><ul><ul><li>The row 0, i.e., the first row of the triangle, contains the single entry 1 . The row 1, i.e., the second row, contains a pair of entries each equal to 1 . </li></ul></ul><ul><ul><li>Calculate the n t h row of the triangle from the preceding row by the following rules: </li></ul></ul>Basic Quantitative Techniques - RVMReddy - ABS
  27. 27. Binomial Coefficients July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  28. 28. Basic Quantitative Techniques - RVMReddy - ABS
  29. 29. Binomial Coefficients <ul><li>The technique known as divide and conquer can be used to compute C(n , r ). </li></ul><ul><li>In the divide-and-conquer technique, a problem is divided into a fixed number, say k , of smaller problems of the same kind. </li></ul><ul><li>Typically, k = 2 . Each of the smaller problems is then divided into k smaller problems of the same kind, and so on, until the smaller problem is reduced to a case in which the solution is easily obtained. </li></ul><ul><li>The solutions of the smaller problems are then put together to obtain the solution of the original problem. </li></ul>July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  30. 30. Thank You Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010

×