12.1 The Fundamental Counting Principal

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12.1The Fundamental Counting Principal
Permutations

• L. Kealii Alicea

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The Fundamental Counting Principal

• If you have 2 events 1 event can occur m ways
  and another event can occur n ways, then the
  number of ways that both can occur is mn
• Event 1 4 types of meats
• Event 2 3 types of bread
• How many diff types of sandwiches can you make?
• 43 12

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3 or more events

• 3 events can occur m, n, p ways, then the
  number of ways all three can occur is mnp
• 4 meats
• 3 cheeses
• 3 breads
• How many different sandwiches can you make?
• 433 36 sandwiches

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• At a restaurant at Cedar Point, you have the
  choice of 8 different entrees, 2 different
  salads, 12 different drinks, 6 different
  deserts.
• How many different dinners (one choice of each)
  can you choose?
• 82126
• 1152 different dinners

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Fund. Counting Principal with repetition

• Ohio Licenses plates have 3 s followed by 3
  letters.
• 1. How many different licenses plates are
  possible if digits and letters can be repeated?
• There are 10 choices for digits and 26 choices
  for letters.
• 101010262626
• 17,576,000 different plates

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How many plates are possible if digits and
numbers cannot be repeated?

• There are still 10 choices for the 1st digit but
  only 9 choices for the 2nd, and 8 for the 3rd.
• For the letters, there are 26 for the first, but
  only 25 for the 2nd and 24 for the 3rd.
• 1098262524
• 11,232,000 plates

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Phone numbers

• How many different 7 digit phone numbers are
  possible if the 1st digit cannot be a 0 or 1?
• 8101010101010
• 8,000,000 different numbers

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Testing

• A multiple choice test has 10 questions with 4
  answers each. How many ways can you complete the
  test?
• 4444444444 410
• 1,048,576

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Using Permutations

• An ordering of n objects is a permutation of the
  objects.

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There are 6 permutations of the letters A, B, C

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA

You can use the Fund. Counting Principal to
determine the number of permutations of n
objects. Like this ABC. There are 3 choices for
1st 2 choices for 2nd 1 choice for 3rd. 321
6 ways to arrange the letters
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In general, the of permutations of n objects is

• n! n(n-1)(n-2)

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12 skiers

• How many different ways can 12 skiers in the
  Olympic finals finish the competition? (if there
  are no ties)
• 12! 121110987654321
• 479,001,600 different ways

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Factorial with a calculator

• Hit math then over, over, over.
• Option 4

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Back to the finals in the Olympic skiing
competition.

• How many different ways can 3 of the skiers
  finish 1st, 2nd, 3rd (gold, silver, bronze)
• Any of the 12 skiers can finish 1st, the any of
  the remaining 11 can finish 2nd, and any of the
  remaining 10 can finish 3rd.
• So the number of ways the skiers can win the
  medals is
• 121110 1320

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Permutation of n objects taken r at a time

• nPr

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Back to the last problem with the skiers

• It can be set up as the number of permutations of
  12 objects taken 3 at a time.
• 12P3 12! 12! (12-3)! 9!
• 121110987654321
  987654321
• 121110 1320

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10 colleges, you want to visit all or some.

• How many ways can you visit
• 6 of them
• Permutation of 10 objects taken 6 at a time
• 10P6 10!/(10-6)! 10!/4!
• 3,628,800/24 151,200

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How many ways can you visitall 10 of them

• 10P10
• 10!/(10-10)!
• 10!/0!
• 10! ( 0! By definition 1)
• 3,628,800

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So far in our problems, we have used distinct
objects.

• If some of the objects are repeated, then some of
  the permutations are not distinguishable.
• There are 6 ways to order the letters M,O,M
• MOM, OMM, MMO
• MOM, OMM, MMO
• Only 3 are distinguishable. 3!/2! 6/2 3

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Permutations with Repetition

• The number of DISTINGUISHABLE permutations of n
  objects where one object is repeated q1 times,
  another is repeated q2 times, and so on
• n! q1! q2!
  qk!

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Find the number of distinguishable permutations
of the letters

• OHIO 4 letters with 0 repeated 2 times
• 4! 24 12
• 2! 2
• MISSISSIPPI 11 letters with I repeated 4 times,
  S repeated 4 times, P repeated 2 times
• 11! 39,916,800 34,650
• 4!4!2! 24242

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Find the number of distinguishable permutations
of the letters

• SUMMER
• 360
• WATERFALL
• 90,720

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A dog has 8 puppies, 3 male and 5 female. How
many birth orders are possible

• 8!/(3!5!)
• 56

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Assignment
12.1 A (1-25) 12.1 B (all)