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Outcomes in the sample space. abc    acb    bac     bca    cab    cba There are six permutations.

Consider the possible arrangements of the letters a , b , and c .  List the outcomes in the sample space. If the order is important, then each arrangement is a permutation of the three letters. Outcomes in the sample space. abc    acb    bac     bca    cab    cba

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Outcomes in the sample space. abc    acb    bac     bca    cab    cba There are six permutations.

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  1. Consider the possible arrangements of the letters a, b, and c.  List the outcomes in the sample space. If the order is important, then each arrangement is a permutation of the three letters Outcomes in the sample space. abc    acb    bac     bca    cab    cba There are six permutations.

  2. Permutation Formula:Use when the item possibilities are alldifferent.n: number of available items or choicesr: the number of items to be selected    Sometimes this formula is written: P(n,r). • Remember: The permutation formula takes order into account. The outcome (red, blue, green) is not the same as the outcome (green, blue, red.)

  3. If n=5 and r=2, what is the value of 5P2? The symbol 5P2 means the number of permutations of 5 items taken 2 at a time.  • A permutation of 5 objects taken 2 at a time is equal to . . .

  4. If there are n items with n1 alike, n2 alike, n3 alike, . . ., nk alike, the number of permutations is calculated by dividing n factorial by the product of the factorials of the number of occurences of each of the like items. Use this permutation formula when some of the item possibilities are alike.

  5. How many permutations are there for the letters in the word, BANANA?First, notice there are six total letters of which the A is repeated 3 times and the N is repeated 2 times. The number of permutations is equal to . . . .  There are six letters in BANANA (6!).The A repeats three times (3!)and the N repeats two times (2!).There are 60 permutations.

  6. How many permutations are there for the letters in the word, HAPPY?Remember to ask yourself, how many total letters are there?  How many letters are repeated? How many times is each letter repeated?  There are five letters in HAPPY (5!).The P repeats two times (2!).There are

  7. 1) Calculate the value of 7P3 A permutation of 7 objects taken 3 ata time is equal to . . .

  8. 2) Calculate the value of 9P4 A permutation of 9 objects taken 4 at a time is equal to . . .

  9. How many permutations are there for the letters in the word, STATISTICS? There are ten letters in STATISTICS (10!). The S repeats three times (3!),the T repeats three times (3!), and the I repeats two times (2!).

  10. When different orderings of the same items are to be counted individually, i.e. each listing represents a different scenario (mn is not the same as nm), the problem involves permutations. When different orderings are not to be counted separately, i.e. the outcome, mn is equivalent to the outcome nm, the problem involves combinations.

  11. Combination Formula:Different orders of the same items are not counted.  The combination formula is equivalent to dividing the corresponding number of permutations by r!.n: number of available items or choicesr: the number of items to be selected    Sometimes this formula is written: C(n,r).

  12. Taking the letters a, b, and c taken two at a time, there are six permutations: {ab, ac, ba, bc, ca, cb}.  If the order of the arrangement is not important, how many of these outcomes are equivalent, i.e. how many combinations are there ab = ba; ac = ca; and bc = cbThe three duplicate permutations would not be counted, therefore three combinations exist

  13. Calculate the value of 7C4. This represents a combination of 7 objectstaken 4 at a time and is equal to

  14. Calculate the value of 9C5 This represents a combination of 9 objects taken 5 at a time and is equal to . . .

  15. Determine whether the following scenarios represent permutations or combinations. Correct answers are provided at the end of this section. 1) Selecting two types of yogurt from the grocery's dairy case from a selection of nine. 2) Selecting your favorite yogurt and then your second favorite yogurt from a selection of nine. 3) Selecting three members from your class to work specific homework problems on the board. 4) Choosing two books to take with you on vacation from the nine books on your shelf. 5) Choosing three CDs to purchase from the music store. 6) Arranging seven photographs on a page of your senior memory book.

  16. 1) Combination2) Permutation3) Permutation4) Combination5) Combination6) Permutation

  17. In how many ways can three class representatives be chosen from a group of twelve students?  If the order of the arrangement is not important, how many outcomes will there be? This represents a combination of 12 objects taken 3 at a time and is equal to 9! 9! 9!

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