Add Maths - F5 - (Version 2007) - Permutation and Combination by nklye

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									               MASTER
                       in




                     &



                       by

                    NgKL
(M.Ed.,B.Sc.Hons.Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH.)
A. PERMUTATIONS

FACTS:

1.   Multiplication Rule:
     If an event A happens in p ways and event B happens in q ways,
     then the number of ways that event A happens followed by event B
     is p x q.

2.   Permutations:
     (a) The number of ways to arrange n unlike objects or events = n!
         where n! = n x (n – 1) x (n – 2) ……3 x 2 x 1.

         Example: 5! = 5 x 4 x 3 x 2 x 1
                     = 120

     (b) 0! = 1

     (c) The number of ways to arrange n objects which have p objects
                     n!
         are alike =
                     p!

         Similarly, if in the n objects have p and q objects which are
         alike, then the number of ways to arrange the n objects =
           n!
          p! q!

     (d) The number of ways to arrange r objects taken out from n
                                     n!
         unlike objects, n P r =
                                 ( n  r )!

                    5P3           5!       5 x 4 x3 x 2 x1
         Example:         =              =                 = 60
                              ( 5  3 )!         2 x1



3.   In a permutations, the order of arrangement of the objects is
     important.

     Example: The arrangement of ABC and CBA are different!


                                                                         2
  Exercise A.1:

1. Calculate the number of ways of       2. Calculate the number of ways of
   forming a mixed-double team in           appointing a male class monitor
   badminton from 5 male and 4              and a female assistant monitor.
   female players.




3. Calculate the number of ways to arrange the following words or digits.

   (a) ADDMATHS                          (b) HIGHSCHOOL




   (c) 3, 4, 9, 9                        (d) 2, 3, 5, 10, 11, 25, 25




4. Find the number of different arrangement of the following;

  (a) Five letters from the words        (b) Four-digits numbers from the
      PROBLEMS, without                      digits 1, 3, 5. 6, 7, 8, 9 if no
      repetitions.                           repetition is allowed.




  (c) Five boys be arranged in a         (d) 3 chairs be arranged in a
      row from a total of 8 boys.            row from 9 different chairs.




                                                                                3
Exercise A.2: Problem Solving I

1. How many 5-digit even numbers         2. How many 4-digit numbers, less
   can be arrange from digits 1, 2, 5,      than 5,000 can be formed from the
   8, 9 if no repetition is allowed?        digits 3, 4, 5, 6 if no repetition is
                                            allowed?




3. How many different arrangements       4. How many different arrangements
   can be formed from the letters of        can be formed from a group of 4
   the word TERBILANG if the                boys and 3 girls, if arrangements
   arrangement begin with a vowel?          begin with a girl?




Exercise A.3: Problem Solving II

1. How many 4-digit numbers can be formed from digits 0 to 9, if the
   numbers are;

   (a) odd?                              (b) greater than 8 000?




2. How many 4-letter word codes can be formed from letters of the word
   HARMONI if the codes;

   (a) contain letter A?                 (b) do not contain any vowel?




                                                                                4
      B. COMBINATIONS

      FACTS:

      1.   The number of combinations of r objects n not alike objects is
                    nC r
           given by

           nC
                 r          n!          n( n  1 )( n  2 ).....( n  r  1 )
                                       
                        ( n  r )! r !                   r!

                           4 C2            4!       4x3x2x1
           Example:               =                          6
                                      ( 4  2 )!2! (2x1)(2x1)
               C0  1
           n
      2.

               Cn  1
           n
      3.

      4.   In a combination, the order of arrangement of objects is not
           important.
           Example: Arrangement of AB and BA are regarded as one
           combination.

Exercise B.1:

1. Find the number of ways of                  2. In how many ways of selecting 3
   choosing 4 letters from the word               brands of hand-phones from 7
   MASTERY.                                       different brands?




3. A computer club has to select 5             4. Four persons have to be chosen
   committee members form 10                      from a group of 4 boys and 3 girls.
   students. Find how many ways of                Find how many ways to select
   selection?                                     them?




                                                                                    5
Exercise B.2: Problem Solving

1. A Parent-Teacher Association has to select 5 teachers from 7 male and 5
   female teachers to be represented in the committee. Determine the
   numbers of ways to select them to the committee if;


   (a) 3 males are to be selected.       (b) At least 2 males to be selected.




2. Four letters are chosen from the word GLORIES. Calculate the number of
   selections if

   (a) the letter G must be selected.    (b) only one vowel to be selected.




3. A café serves 5 types of food and 3 types of drink for breakfast. A customer
   can choose from 1 to 3 types of food or drink for a fixed price. Find the
   number of different choices the customer can make with that price if;


   (a) only food are to be chosen.       (b) only one type of drink must be
                                             chosen.




                                                                                6
       TUTORIAL

 1.   Diagram 1 shows seven cards.

         U        N       I       F        O       R        M

                              DIAGRAM 1

      A four-letter code is to be formed using four of these cards. Find,
      (a) the number of different four-letter codes that can be formed,
      (b) the number of different four-letter codes which end with a
          consonant.
                                                                        [4 marks]




                                      Answer: (a) …………………………………..

                                               (b)……………………………………



2.    A debating team consists of 5 students. These 5 students are chosen
      from 4 monitors, 2 assistant monitors and 6 prefects. Calculate the
      number of different ways the team can be formed if
      (a) there is no restriction,
      (b) the team contains only 1 monitor and exactly 3 prefects. [4 marks]




                                      Answer: (a) …………………………………..

                                               (b)……………………………………
                                                                               7
Quadran II
Sine positive
 (180o )
      3.   Diagram 2 shows five cards of different letters.

                         M       A        T          H        S

                                     DIAGRAM 2

           (a) Find the number of possible arrangements, in a row, of all the cards.
           (b) Find the number of these arrangements in which the letters A and H
               are side by side.                                           [4 marks]




                                              Answer: (a) …………………………………..

                                                         (b)……………………………………



      4.   Diagram 3 shows 5 letters and 3 digits.

                           A    B     C   D    E     6    7       8

                                      DIAGRAM 3

           A code is to be formed using those letters and digits. The code must
           consists of 3 letters followed by 2 digits.
           How many codes can be formed if no letter or digit is repeated in each
           code?                                                       [3 marks]




                                              Answer: ….…………………………………..

                                                                                       8

								
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