What are we doing today?

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What are we doing today?

• Need a calculator
• Notes Basic Combinatorics
• Go over quiz
• Homework

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Definitions

• Independent Events
• the outcome of one event does not affect the
  outcome of any other event.
• Dependent Events
• the outcome of one event does affect the outcome
  of another event.

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Basic Counting Principle

• Suppose one event can be chosen in p different
  ways and another independent event can be chosen
  in q different ways. Then the two events can be
  chosen successively in pq ways.
• This can be extended to any number of events,
  just multiply the number of choices for each
  event.

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Example

• How many sundaes are possible if you can only
  choose one from each of the following categories?
• ice cream flavors chocolate, vanilla,
  strawberry, rocky road
• sauce hot fudge, caramel
• toppings cherries, whipped cream, sprinkles
• (4)(2)(3) 24 different sundaes

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Example

• How many different license plates can be made if
  each plate consists of 2 digits followed by 3
  letters followed by 1 digit?
• Unless told otherwise, always assume all letters
  of the alphabet, all digits 0-9, and repetition
  is allowed. Treat each space as an event.
• (10)(10)(26)(26)(26)(10)
• 17,576,000 possible combinations.

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Example

• A test consists of 8 multiple choice questions.
  How many ways can the 8 questions be answered if
  each question has 4 possible answers?
• (4)(4)(4)(4)(4)(4)(4)(4)
• 48
• 65,536

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Factorials

• n!
• definition product of consecutive numbers from 1
  to n.
• Example 8! (8)(7)(6)(5)(4)(3)(2)(1)
• 40,320

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Permutations

• An arrangement of objects in a specific order or
  selecting all of the objects.
• The number of permutations of n objects taken r
  at a time, denoted P(n,r) or nPr, is

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Example
There are ten drivers in a race. How many
outcomes of first, second, and third place are
possible?
720 ways
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Example
There are 30 students in the Art Club, how many
ways can the club select the President, Vice
President, and Secretary for the club? 30P3
24,360 ways
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Combinations

• An arrangement of objects in which order does not
  matter.
• Difference between permutations and combinations
• Combinations grouping of objects
• Permutation putting objects in specific places
  or positions, or selecting all of the objects.

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Permutations vs Combinations

• Select a committee of 5 people from a group of 33
  people.
• Combination (order doesnt matter)
• Elect a President, Vice President, Treasurer,
  Secretary from a group of 40 people.
• Permutation (putting in specific places)
• Pick your favorite soda, and your second favorite
  soda from a group of 8 sodas.
• Permutation (putting in specific places)
• Buy 3 types of soda at Giant from a group of 30
  sodas.
• Combination (order doesnt matter)
• Arrange the entire set of 12 books on a shelf.
• Permutation (arranging all the objects)

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Example

• In a study hall of 20 students, the teacher can
  send only 6 to the library. How many ways can
  the teacher send 6 students?

38,760 ways
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Example

• Jessie is at the library and wants to sign out 8
  books but she can only sign out 3. How many ways
  can she choose which books to sign out?

56 ways
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Counting Subsets of an n-set

• A local pizza shop offers patrons any combination
  of up to 10 different toppings. How many
  different pizzas can be ordered if patrons can
  choose any number of toppings (0 through 10)?
• Each topping can be seen as a yes or no question
  so each has 2 options
• 210 1024 different pizzas are possible.
• (This includes no toppings and picking all
  toppings)

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Homework

• Pg 708 1-8, 11-22
• Bring your textbook tomorrow.
• SAT Math II take end of this year