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Activity 6 - 7

Activity 6 - 7. Selecting & Rearranging Things. Objectives. Determine the number of permutations Determine the number of combinations Recognize patterns modeled by counting techniques. Vocabulary. Factorial – n! is defined to be n! = n∙(n-1)∙(n-2)∙(n-3)….. (3)∙(2)∙(1)

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Activity 6 - 7

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  1. Activity 6 - 7 Selecting & Rearranging Things

  2. Objectives • Determine the number of permutations • Determine the number of combinations • Recognize patterns modeled by counting techniques

  3. Vocabulary • Factorial – n! is defined to be n! = n∙(n-1)∙(n-2)∙(n-3)….. (3)∙(2)∙(1) • Permutation – is an arrangement of objects, from first to last, where the order in which the objects are selected is most important. The symbol nPr represents the number of permutations of r objects selected from n objects. • Combination – is a collection of objects from larger collections, where the order of selection is not important. The symbol nCr represents the number of combinations of n objects taken r at a time.

  4. Activity In working through the previous activities, you may have noticed there are many different varieties of counting problems. Sometimes a tree or list may be feasible to display all possibilities. But usually the overwhelming number of possibilities makes constructing a list or tree impractical. In this activity you will explore two more types of counting problems, permutations and combinations – both of which use the multiplication principle of counting. You have 5 letters in your name and wish to arrange them on your paper. In how many ways can this be done? 5  4  3  2  1 = 120

  5. Factorials A factorial, number!, is a series of numbers (from the number to 1) multiplied together. 5! = 54321 = 120 7! = 7654321 = 5040 n! = n  (n – 1)  (n – 2)  (n – 3)  …  1 Our calculation has a factorial (!) key under MATH, PRB type the number in and hit MATH and select PRB, 4

  6. Example 1 Find the following factorials: • 6! • 10! • 8!---- 6! = 6  5! = 6  120 = 720 = 10  9  8  7  6! = 5040  720 = 3,628,800 8  7  6! 8  7 = -------------- = ------- = 56 6! 1

  7. Activity cont What if we didn’t order the entire collection, but rather used a subset of that collection. There are 26 letters in the alphabet. How many 5 letter passwords (can’t use the same letter more than once) can be made out of the 26 letters? 26  25  24  23  22 = 7,893,6000 This is called a permutation of 26 items taken 5 at a time.

  8. Permutations Number of Permutations of n Distinct Objects taken r at a time: N objects are distinct Once used an object cannot be repeated Order is important n! nPr = ----------- (n – r)! Type in n, hit MATH; select PRB, 2 and type in r; ENTER

  9. Example 2 Find the following permutations: • 7P3 • 9P5 • 6P4 n! 7! 7654! nPr = ----------- = -------- = -------------- = 210 (n – r)! 4! 4! n! 9! nPr = ----------- = -------- = 15120 (n – r)! 5! n! 6! nPr = ----------- = -------- = 30 (n – r)! 4!

  10. Example 3 In a horse racing “Trifecta”, a gambler must pick which horse comes in first, which second, and which third. If there are 8 horses in the race, and every order of finish is equally likely, what is the chance that any ticket is a winning ticket? The probability that any one ticket is a winning ticket is 1 out of 8P3, or 1 out of 336

  11. Activity cont What if order was unimportant to us? What if only being selected was the most important thing? There are 26 students in class; 5 students will be chosen to go on a trip to visit the set of National Treasure 3 filming in Richmond next summer. How many different combinations of students can be done? 26P5= 7,893,600 permutations of 5 students and 5P5 = 120 permutations for each group of 5 Since order is unimportant: 26P5 / 5P5 = 65,780 This is called a combination of 26 items taken 5 at a time.

  12. Combinations Number of Combinations of n Distinct Objects taken r at a time: N objects are distinct Once used an object cannot be repeated (no repetition) Order is not important n! nPr nCr = ----------- = ---------- r!(n – r)! rPr Type in n, hit MATH; select PRB, 3 and type in r; ENTER

  13. Example 4 Find the following combinations: • 7C3 • 9C5 • 6C4 n! 7! 765 nCr = ----------- = --------- = ---------- = 35 r!(n – r)! 3! 4! 321 n! 9! nCr = ----------- = --------- = 126 r!(n – r)! 5! 4! n! 6! nCr = ----------- = --------- = 15 r!(n – r)! 4! 2!

  14. Example 5 If there are 8 researchers and 3 of them are to be chosen to go to a meeting, how many different groupings can be chosen?

  15. Permutation vs Combination • Comparing the description of a permutation with the description of a combination • The only difference is whether order matters

  16. How to Tell Is a problem a permutation or a combination? • One way to tell • Write down one possible solution (i.e. Roger, Rick, Randy) • Switch the order of two of the elements (i.e. Rick, Roger, Randy) • Is this the same result? • If no – this is a permutation – order matters • If yes – this is a combination – order does not matter

  17. Summary and Homework • Summary • Permutations are arrangement of objects where order selected is the most important • Combinations are collections of objects where order selected is not important • Order is the difference between permutations and combinations • Homework • pg 762 – 763; problems 1-4, 8, 10 n! nPr = ----------- (n – r)! n! nCr = ----------- r!(n – r)!

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