Permutations and Combinations

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Permutations and Combinations

• Homework Permutation and Combinations WS

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WARM UP

• There are 7 green marbles, 4 red marbles, and 2
  blue marbles in the bag. Jenny picked a green
  marble from the bag, without replacement. What
  is the probability that the next marble picked is
  also green?

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WARM UP- SOLUTION

• There are 7 green marbles, 4 red marbles, and 2
  blue marbles in the bag. Jenny picked a green
  marble from the bag, without replacement. What
  is the probability that the next marble picked is
  also green?
• 6/12 or 1/2

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COMBINATIONS

• Combination- order doesnt matter.
• If you are dealt 5 cards from a deck it doesnt
  matter what order you get them, when you pick up
  your hand you have 1 combination of cards.
• A combination is a grouping of the elements from
  a set in which the order doesnt matter.
• In a combination, abc and acb would be considered
  the same The elements are the same in both
  groups, and the order in which they appear does
  not matter.

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EXAMPLE 1

• How many combinations are there of the letters
  a, b, c and d using all letters?
• How many combinations are there using 3 of the
  letters?

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EXAMPLE 1- SOLUTIONS

• How many combinations are there of the letters
  a, b, c and d using all letters?
• There is 1 combination.
• How many combinations are there using 3 of the
  letters?
• abc, abd, acd, bcd
• There are 4 combinations of 4 letters taken 3 at
  a time.

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EXAMPLE 2

• How many combinations are there of the 4 letters
  a, b, c and d using 2 letters at a time?

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EXAMPLE 2- SOLUTION

• How many combinations are there of the 4 letters
  a, b, c and d using 2 letters at a time?
• ab ac ad bc bd cd
• There are 6 combinations.

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EXAMPLE 3

• How many combinations are there of the 4 letters
  a, b, c and d using 1 letter at a time?

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EXAMPLE 3- SOLUTION

• How many combinations are there of the 4 letters
  a, b, c and d using 1 letter at a time?
• a b c d
• There are 4 combinations.

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COMBINATIONS- FORMULA

• The combination of n things taken r at a time is
• 5! is read five factorial.
• It means (5)(4)(3)(2)(1) 120

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EXAMPLE 4

• Find 10C6
• There are lots of factors that you can cross out
  once you expand your factorials.

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EXAMPLE 5

• Find 6C2 ,9C4 and 10C7.
• 6C2
• 9C4
• 10C7

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EXAMPLE 5- SOLUTIONS

• Find 6C2 ,9C4 and 10C7.
• 6C2
• 9C4
• 10C7

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EXAMPLE 6

• There are 6 questions on Elizabeths essay test.
  She only needs to answer 2 of them, she can
  choose any 2 that she wants. How many different
  combinations of 2 test questions can Elizabeth
  answer?

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EXAMPLE 6-SOLUTION

• There are 6 questions on Elizabeths essay test.
  She only needs to answer 2 of them, she can
  choose any 2 that she wants. How many different
  combinations of 2 test questions can Elizabeth
  answer?
• 6C2

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PERMUTATIONS

• A permutation is an arrangement of objects in an
  specific order.
• Order matters.
• 125 is very different that 512

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EXAMPLE 7

• How many permutations are there using the letters
  ABC?

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EXAMPLE 7- SOLUTIONS

• How many permutations are there using the letters
  ABC?
• ABC, ACB, BCA, CBA, BCA, BAC 6
• These are dependent events, and using the
  fundamental counting principle we get
• 3 x 2 x 1 or 3!

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PERMUTATIONS- FORMULA

• The permutations of n things taken r at a time is

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EXAMPLE 8

• Find 6P2 ,9P4 and 8P5.
• 6P2
• 9P4
• 8P5

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EXAMPLE 8- SOLUTIONS

• Find 6P2 ,9P4 and 8P5.
• 6P2
• 9P4
• 8P5

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EXAMPLE 9

• Determine if each is a permutation or a
  combination.
• Assuming that any arrangement of letters forms a
  'word', how many 'words' of any length can be
  formed from the letters of the word MATH?
• Find the number of ways to take 20 objects and
  arrange them in groups of 5 at a time where order
  does not matter.
• How many ways are there to select a subcommittee
  of 7 members from among a committee of 17??

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EXAMPLES-SOLUTIONS

• Determine if each is a permutation or a
  combination.
• Assuming that any arrangement of letters forms a
  'word', how many 'words' of any length can be
  formed from the letters of the word MATH?
• Permutation
• Find the number of ways to take 20 objects and
  arrange them in groups of 5 at a time where order
  does not matter.
• Combination
• How many ways are there to select a subcommittee
  of 7 members from among a committee of 17?
• Combination