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Counting Fundamentals

Counting Fundamentals. Ginger Holmes Rowell, Middle TN State University MSP Workshop June 2006. Objectives.

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Counting Fundamentals

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  1. Counting Fundamentals Ginger Holmes Rowell, Middle TN State University MSP Workshop June 2006

  2. Objectives • Solve basic problems that involve counting techniques, including the multiplication principle, permutations, and combinations; use counting techniques to understand various situations (e.g., number of ways to order a set of objects, to choose a subcommittee from a committee, to visit n cities)

  3. Overview • Fundamentals of Counting • Multiplication Principal • Permutations: ordered arrangements • Combinations: unordered arrangements • Selected Activities

  4. Counting Techniques • Fundamentals of Counting • Permutations: ordered arrangements • Combinations: unordered arrangements

  5. Fundamentals of Counting • Q: Jill has 9 shirts and 4 pairs of pants. How many different outfits does she have? • A: 9 x 4 = 36 36 different outfits

  6. Multiplication Principle: If there are a ways of choosing one thing, and b ways of choosing a second thing after the first is chosen, then the total number of choice patterns is: a x b Fundamentals of Counting

  7. Q: 3 freshman, 4 sophomores, 5 juniors, and 2 seniorsare running for SGA representative. One individualwill be selected from each class. How many different representative orderings are possible? A: 3 · 4 · 5 · 2 = 120 different representative orderings Fundamentals of Counting

  8. If there are a ways of choosing one thing, b ways of choosing a second thing after the first is chosen, and c ways of choosing a third thing after the first two have been chosen…and z ways of choosing the last item after the earlier choices, then the total number of choice patterns is a x b x c x … x z Generalized Multiplication Principle

  9. Example • Q: When I lived in Madison Co., AL, the license plates had 2 fixed numbers, 2 variable letters and 3 variable numbers. How many different license plates were possible? • A: 26 x 26 x 10 x 10 x 10 = 676,000 different plates

  10. Fundamentals of Counting Q: How many more license plate numbers will Madison County gain by changing to 3 letters and 2 numbers? A: 26 x 26 x 26 x 10 x 10 = 1,757,600 • 1,757,600 – 676,000 = 1,081,600 more license plate numbers

  11. Permutations: Ordered Arrangements • Q: Given 6 people and 6 chairs in a line, how many seating arrangements (orderings) are possible? • A: 6 · 5 · 4 · 3 · 2 · 1 = 6! = 720 orderings

  12. Permutations: Ordered Arrangements • Q: Given 6 people and 4 chairs in a line, how many different orderings are possible? • A: 6 · 5 · 4 · 3 = 360 different orderings

  13. Permutations:Ordered Arrangements • Permutation of n objects taken r at a time: r-permutation, P(n,r), nPr Q: Given 6 people and 5 chairs in a line, how many different orderings are possible? A: 6 · 5 · 4 · 3 · 2 = 720 different orderings

  14. Permutations:Ordered Arrangements nPr = n(n-1)···(n-(r-1)) = n(n-1)···(n-r+1) = n(n-1)···(n-r+1) (n-r)! (n-r)! = n(n-1)···(n-r+1)(n-r)···(3)(2)(1) (n-r)! = n! (n-r)!

  15. Permutations:Ordered Arrangements • Q: How many different batting orders are possible for a baseball team consisting of 9 players? • A: 9 · 8 · 7 ··· 3 · 2 · 1 = 9! = 362,880 batting orders • Note: this is equivalent to 9P9. 9P9 = 9! = 9! = 9! (9-9)! 0!

  16. Permutations:Ordered Arrangements • Q: How many different batting orders are possible for the leading four batters? • A: 9 · 8 · 7 · 6 = 3,024 orders 9P4 = 9! = 9! = 9! (9-4)! 5! 5!

  17. Permutations:Indistinguishable Objects • Q: How many different letter arrangements can be formed using the letters T E N N E S S E E ? • A: There are 9! Permutations of the letters T E N N E S S E E if the letters are distinguishable. • However, 4 E’s are indistinguishable. There are 4! ways to order the E’s.

  18. Permutations:Indistinguishable Objects, Cont. • 2 S’s and 2 N’s are indistinguishable. There are 2! orderings of each. • Once all letters are ordered, there is only one place for the T. • If the E’s, N’s, & S’s are indistinguishableamong themselves, then there are 9! = 3,780 different orderings of(4!·2!·2!) T E N N E S S E E

  19. Permutations:Indistinguishable Objects Subsets of Indistinguishable Objects Given n objects of which a are alike, b are alike, …, and z are alike There are n! permutations. a!·b!···z!

  20. Combinations:Unordered Arrangements • Combinations: number of different groups of size r that can be chosen from a set of n objects (order is irrelevant) • Q: From a group of 6 people, select 4. How many different possibilities are there? • A: There are 6P4=360 different orderings of 4 people out of 6. 6·5·4·3 = 360 = 6P4 = n! (n -r)!

  21. Unordered Example continued • However the order of the chosen 4 people is irrelevant. There are 24 different orderings of 4 objects. 4 · 3 · 2 · 1 = 24 = 4! =r! • Divide the total number of orderings by the number of orderings of the 4 chosen people.360 = 15 different groups of 4 people. 24

  22. Combinations: Unordered Arrangements The number of ways to choose r objects from a group of n objects. C(n,r) or nCr, read as “n choose r”

  23. Combinations: Unordered Arrangements • Q: From a group of 20 people, a committee of 3 is to be chosen. How many different committees are possible? • A:

  24. Combinations: Unordered Arrangements • Q: From a group of 5 men & 7 women, how many different committees of 2 men & 3 women can be found? • A: There are 5C2 groups of men & 7C3 groups of women. Using the multiplication principle

  25. Practice Problem • You have 30 students in your class, which will be arranged in 5 rows of 6 people. Assume that any student can sit in any seat. • How many different seating charts could you have for the first row? • How many different seating charts could you have for the whole class?

  26. It’s Your Turn • Make up three counting problems which would interest your students, include one permutation and one combination and one of your choice. • Calculate the answer for these problems.

  27. Questions ??? • Fundamentals of Counting? • Permutations? • Combinations?

  28. Homework

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