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Permutations of
                     Non-Distinguishable
                        Objects ... like
     but for real
                          twins. ;-)
     this time ...




My Lovely Twins
Permutations (the quot;Pickquot; Formula)
A permutation is an ordered arrangement of objects.
                     n is the number of objects
                                                      On the calculator ...
                     available to be arranged
                                                       Press: [MATH]
                                                               [<] (Prb)
                     r is the number of objects
                                                                  [2] (nPr)
                     that are being arranged.
  Examples:
In how many ways can 5 people          In how many ways can six
be seated in a straight line?          students be seated in 8 vacant
                                       seats?
(a) How many “words” of 4 different letters each can be made from the
letters A, E, I, O, R, S, T?
                                  HOMEWORK


(b) How many of these words begin with a vowel and end with a
consonant?




(c) In how many of these words do vowels and consonants alternate?
(a) How many numbers of 5 different digits each can be formed from
                                            HOMEWORK
the digits 0, 1, 2, 3, 4, 5, 6?




(b) How many of these numbers are even?




 (c) What is the probability that one of these numbers is even if the
 digits are randomly chosen ?
(a) In how many ways can 4 English books and 3 French books be
                                HOMEWORK
arranged in a row on a shelf?



 (b) In how many of these ways will the French books be together?




(c) What is the probability the French books will be together?
If a fair coin is tossed 4 times, what is the probability of obtaining
exactly 2 heads?
T

T
    T
        T
        T
    T
Permutations of non-distinguishable objects ...
The number of ways to arrange n objects that contain k1, k , k 3, ... sets of
                                                          2
non-distinguishable objects is given by:




 Examples:

Find the number of different quot;wordsquot; that can be made by rearranging
the letters in the word:
      (a) BOOK                            (b) MISSISSIPPI
Applied 40S March 10, 2009
If a fair coin is tossed 4 times, what is the probability of obtaining
exactly 2 heads?
If a fair coin is tossed 4 times, what is the probability of obtaining
exactly 2 heads?
All the letters of the word MANITOBA are arranged at random in a
                                                HOMEWORK
 row. How many ways can this be done?




How many arrangements will have the two A’s next to each other?




What is the probability that this random arrangement will have the two
A’s next to each other?
(a) In how many ways can the letters of the word GEOMETRY be
                                                      HOMEWORK
arranged so that vowels and consonants alternate?




(b) In how many of these ways is Y the last letter?




 (c) If one of these quot;wordsquot; is randomly selected, what is the probability
 that Y is the last letter?

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Applied 40S March 10, 2009

  • 1. Permutations of Non-Distinguishable Objects ... like but for real twins. ;-) this time ... My Lovely Twins
  • 2. Permutations (the quot;Pickquot; Formula) A permutation is an ordered arrangement of objects. n is the number of objects On the calculator ... available to be arranged Press: [MATH] [<] (Prb) r is the number of objects [2] (nPr) that are being arranged. Examples: In how many ways can 5 people In how many ways can six be seated in a straight line? students be seated in 8 vacant seats?
  • 3. (a) How many “words” of 4 different letters each can be made from the letters A, E, I, O, R, S, T? HOMEWORK (b) How many of these words begin with a vowel and end with a consonant? (c) In how many of these words do vowels and consonants alternate?
  • 4. (a) How many numbers of 5 different digits each can be formed from HOMEWORK the digits 0, 1, 2, 3, 4, 5, 6? (b) How many of these numbers are even? (c) What is the probability that one of these numbers is even if the digits are randomly chosen ?
  • 5. (a) In how many ways can 4 English books and 3 French books be HOMEWORK arranged in a row on a shelf? (b) In how many of these ways will the French books be together? (c) What is the probability the French books will be together?
  • 6. If a fair coin is tossed 4 times, what is the probability of obtaining exactly 2 heads?
  • 7. T T T T T T
  • 8. Permutations of non-distinguishable objects ... The number of ways to arrange n objects that contain k1, k , k 3, ... sets of 2 non-distinguishable objects is given by: Examples: Find the number of different quot;wordsquot; that can be made by rearranging the letters in the word: (a) BOOK (b) MISSISSIPPI
  • 10. If a fair coin is tossed 4 times, what is the probability of obtaining exactly 2 heads?
  • 11. If a fair coin is tossed 4 times, what is the probability of obtaining exactly 2 heads?
  • 12. All the letters of the word MANITOBA are arranged at random in a HOMEWORK row. How many ways can this be done? How many arrangements will have the two A’s next to each other? What is the probability that this random arrangement will have the two A’s next to each other?
  • 13. (a) In how many ways can the letters of the word GEOMETRY be HOMEWORK arranged so that vowels and consonants alternate? (b) In how many of these ways is Y the last letter? (c) If one of these quot;wordsquot; is randomly selected, what is the probability that Y is the last letter?