12-1 and 12-2

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12-1 and 12-2

• Permutations and Combinations
• Get out that calculator!! ?

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If we were to have a class trip to Great
Adventure

• and there were only 3 rides (Ferris Wheel, Roller
  Coaster and Water Ride), in how many different
  orders could you ride them?
• 6 paths, right? You have 3 to choose from then
  2, then one.
• (3 rides, picking one at a time)

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Fundamental Principle of Counting

• If there are m1 ways to do a first task, m2 ways
  to do a second task, m3 ways to do a third task
  mn ways to do the nth task,
• then the total possible patterns or ways you
  could do all the tasks is
• m1m2m3...mn

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Examples

1. How many 3 digit numbers can be formed if
   repetitions are allowed?
2. How many 3 digit numbers can be formed if
   repetitions are NOT allowed?
3. How many 4 letter words can be made if
   duplication of letters is allowed?
4. How many 4 letter words can be made if
   duplication of letters is not allowed?

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Examples

• How many 3 letter words can be formed in the
  first letter must be q? (no repetitions)
• How many 3 letter words can be formed with middle
  letter m if repetitions is allowed?
• 7. In a game of poker, how many possible 5 card
  hands are there?

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A new notation

• ! is called the factorial symbol
• n! n(n-1)(n-2)(n-3)..1
• 3! 3(2)(1) 6
• 5! 5(4)(3)(2)(1) 120

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Permutations

• A permutation of n elements taken r at a time
  is an ordered arrangement (without repetition) of
  r of the n elements and it is called nPr.
• The thing to remember is that ORDER MATTERS!!

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Examples

• 8. How many ways can 4 coins (dime, nickel,
  quarter and penny) be arranged in a row?
• 9. There is a club of 5 boys and 8 girls. The
  president will be a girl, the VP will be a boy,
  and the secretary and treasurer can each be
  either a boy or a girl. How many councils are
  possible??

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What if order doesnt matter?

• What if everyone in a class of 16 had to shake
  hands. Using the fundamental theorem of
  counting, dont you get twice the number of
  handshakes? Because we would divide 16P2 by 2
  since order doesnt matter.
• When order doesnt matter, it is called a
• Combination

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Consider this question

• You and 3 of your friends go to Martinsville
  Pizza for an afternoon snack. You decide to get
  a large pie. Martinsville pizza offers 5
  possible toppings (pepperoni, mushroom, sausage,
  green peppers and onion). How many possible
  pizza orders could be made?
• Keep in mind
• You may have a pizza from 1 to 5 toppings.

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Some shorthand

• P pepperoni
• M mushroom
• S sausage
• G green peppers
• O onions

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One topping
No topping
1 no topping pizza

• P or M or S or G or O 5 one topping pizzas

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Two topping

• (Note Sausage and Mushroom is the same as
  Mushroom and Sausage, right?)
• P with M, S, G or O (4)
• M with S, GP, or O (3)
• S with GP or O (2)
• GP with O (1)
• 10 two topping pizzas

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Three topping

• P with M,S or M,G or M,O
• with S,G or S,O
• with G,O (6)
• M with S,G or S,O
• with G,O (3)
• S with G,O (1)
• 10 three topping pizzas

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Four topping

• P with M,S,G or M,S,O or M,G,O or S,G,O
• 
  (4)
• M with S,G,O
  (1)
• 5 four topping pizzas

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Five topping

• Theres only one
• P, M, S, G and O.
• 1 Five topping pizzas

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So how many possible pizzas is this?

• 15101051 32 possible pizzas.
• Is there an easier way to determine these numbers
  without actually writing out the possibilities?
  This method seems very tedious what if you are
  at an ice cream store and have 12 ice cream
  flavors and 9 toppings?

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Lets just see if we can determine a formula

• Look at the 2 topping pizzas.
• At first, how many toppings can you choose from?
  
• How about the next number of choices?
• Look at this to see if it makes sense

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P M S G O
M S G O P S G O P M G O
P M S O P M S G
This looks like 20, which is 5 times 4. Why did
we only have 10? Because we cancelled the
overlap. There were twice as many pizzas
because there were 2 sets of everything. Pepperoni
/Onion is the same as Onion/Pepperoni.
20

• Since there were two elements, for each
  combination, there was the matching one (in
  reverse order). So by dividing by the overlap,
  we got the true number of possibilities.
• What if there were 10 possible toppings and we
  were looking at two topping pizzas?
• 109 90 then divide by 2, so 45 possible pizzas.

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What about other numbers of toppings?

• What if you did 3 toppings as if order mattered?
  
• What was the actual number of 3 topping pizzas?
• What if you did 4 toppings as if order mattered?
  
• What was the actual number of 4 topping pizzas?
• So what must we divide 5P3 by?
• What must we divide 5P4 by?

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A Combination

• A Combination n elements, r at a time, is given
  by the symbol
• And can be expanded as

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What does the formula do?

• determines the total number of
• possibilities (including overlap)
• and dividing by r! takes care of the overlaps.
• 2! 2 (with 2 toppings)
• 3! 6 (with 3 toppings)

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Examples

• 10. I have a half dollar, penny, nickel, dime
  and quarter. How many 3 coin combinations are
  there?
• 11. I have a group of 10 boys, 15 girls. How
  many committees of 5 can I create of 2 boys and 3
  girls?
• 12. How many teams of 6 hockey players be chosen
  from a group of 12 if position (i.e. order)
  doesnt matter?

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Homework

• 745 11-25 odd
• 752 11-17 odd