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Permutations and Combinations

Permutations And Combinations PDF Notes, Important Questions and Synopsis

 

SYNOPSIS

  1. Fundamental Principle of Counting
    • Permutation is the number of ways to arrange things.
      Eg: Password is 2045
      (order matters)

    • It is denoted by P(n, r) and given by
      P(n, r) =begin mathsize 12px style straight P presuperscript straight n subscript straight r equals fraction numerator straight n factorial over denominator open parentheses straight n minus straight r close parentheses factorial end fraction end style, where 0 ≤ r ≤ n
      n →
        number of things to choose from
      r →
       number of things we choose
      ! →
       factorial
    • Combination is the number of ways to choose things.Eg: A cake contains chocolates, biscuits, oranges and cookies.
      (Order does not matter)
    • It is denoted by C(n, r) and given by
      C(n, r) =begin mathsize 12px style straight C presuperscript straight n subscript straight r equals fraction numerator straight n factorial over denominator straight r factorial left parenthesis straight n minus straight r right parenthesis factorial end fraction equals open parentheses table row straight n row straight r end table close parentheses end style, where 0 ≤  
        number of things to choose from
      →  number of things we choose
      →  factorial
  2. Permutation
    If n is the number of distinct things and r things are chosen at a time.
    i. Permutation of objects when all are not distinct:
       Permutation =begin mathsize 12px style fraction numerator straight n factorial over denominator straight P subscript 1 factorial straight P subscript 2 factorial. .... straight P subscript straight r factorial end fraction end style , begin mathsize 12px style straight P subscript straight r end style Number of things among ‘n’ which are alike of rth type.
    ii. Circular permutation

    1. When clockwise and anti-clockwise arrangements are different:
        Number of permutations: (n - 1)!
    2. When clockwise and anti-clockwise arrangements are the same:
        Number of permutations: begin mathsize 12px style 1 half left parenthesis straight n minus 1 right parenthesis factorial end style
    iii. Permutation under restrictions
         Selecting and arranging r distinct objects from n
  1.  When ‘k’ particular things are always to be included.
    Number of permutations:
    begin mathsize 12px style fraction numerator left parenthesis straight n minus straight k right parenthesis factorial straight r factorial over denominator left parenthesis straight n minus straight r right parenthesis factorial left parenthesis straight n minus straight k right parenthesis factorial end fraction end style
  2. When a particular thing is always to be included (k = 1).
    Number of permutations:
    begin mathsize 12px style fraction numerator left parenthesis straight n minus 1 right parenthesis factorial straight r factorial over denominator left parenthesis straight n minus straight r right parenthesis factorial left parenthesis straight n minus 1 right parenthesis factorial end fraction end style
  3. When ‘k’ particular things are never included.
    Number of permutations:
    begin mathsize 12px style fraction numerator left parenthesis straight n minus straight k right parenthesis factorial over denominator left parenthesis straight n minus straight k minus straight r right parenthesis factorial end fraction end style
  4.  When a particular thing is never included.
    Number of permutations:
    begin mathsize 12px style fraction numerator left parenthesis straight n minus 1 right parenthesis factorial over denominator left parenthesis straight n minus straight r minus 1 right parenthesis factorial end fraction end style
  5. When ‘l’ particular things always come together.
    Number of permutations: (n - l + 1)!  l!
  6. When ‘l’ particular things never come together.
    Number of permutations: n! - (n - l + 1)!  l!
 i. Permutations with repetition Number of permutations: nPr= nr (Repetition, order matters)

    Out of          ←
  Taking 2 at a time
  
ii.  Permutations without repetition
begin mathsize 12px style blank to the power of straight n straight P subscript straight r equals fraction numerator straight n factorial over denominator open parentheses straight n minus straight r close parentheses factorial end fraction end style
(No repetition, order matters)
Out of              ←  Taking 2 at a time
 
3.Combination


i Total number of combinations

  1. Ways of selecting one or more things at a time.
    Number of combinations:begin mathsize 12px style straight C presuperscript straight n subscript 1 plus straight C presuperscript straight n subscript 2 plus. .. plus straight C presuperscript straight n subscript straight n equals 2 to the power of straight n minus 1 end exponent end style
  2. When ‘k1’ alike objects of one kind, ‘k2’ alike objects of the second kind … ‘kn’ alike
    objects of the nth kind.
    Number of combinations:begin mathsize 12px style left parenthesis straight k subscript 1 plus 1 right parenthesis left parenthesis straight k subscript 2 plus 1 right parenthesis. ... left parenthesis straight k subscript straight n plus 1 right parenthesis minus 1 end style
  3. When ‘k1’ alike objects of one kind, ‘k2’ alike objects of the second kind … ‘kn’ alike
    objects of the nth kind and rest ‘p’ elements are different.
    Number of combinations:begin mathsize 12px style open square brackets left parenthesis straight k subscript 1 plus 1 right parenthesis left parenthesis straight k subscript 2 plus 1 right parenthesis. ... left parenthesis straight k subscript straight n plus 1 right parenthesis close square brackets 2 to the power of straight p minus 1 end style
 ii  Combinations under restriction
  1. When ‘k’ particular things are always to be included
    Number of combinations:begin mathsize 12px style straight C presuperscript left parenthesis straight n minus straight k right parenthesis end presuperscript subscript left parenthesis straight r minus straight k right parenthesis end subscript end style
  2. When a particular thing is always to be included (k = 1)
    Number of combinations:begin mathsize 12px style straight C presuperscript left parenthesis straight n minus 1 right parenthesis end presuperscript subscript left parenthesis straight r minus 1 right parenthesis end subscript end style
  3. When ‘k’ particular things are never included
    Number of combinations:begin mathsize 12px style straight C presuperscript left parenthesis straight n minus straight k right parenthesis end presuperscript subscript straight r end style
  4. When ‘k’ particular things never come together
    Number of combinations:begin mathsize 12px style straight C presuperscript straight n subscript straight r minus straight C presuperscript left parenthesis straight n minus straight k right parenthesis end presuperscript subscript left parenthesis straight r minus straight k right parenthesis end subscript end style
 i. Combination with repetition
  Formula:begin mathsize 12px style straight C presuperscript open parentheses straight n plus straight r minus 1 close parentheses end presuperscript subscript straight r end style  (Repetition, order does not matter)
  Out of      Taking 2 at a time

  
  ii. Combination without repetition
  Formula: begin mathsize 12px style fraction numerator straight n factorial over denominator open parentheses straight n minus straight r close parentheses factorial end fraction end style
  (No repetition, order does not matter)
  Out of     Taking 2 at a time
  

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