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Unit: Probability 6-7: Permutations and Combinations. Essential Question: How is a combination different from a permutation?. 6-7: Combinations and Permutations. Permutation: an arrangement of items in a particular order

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unit probability 6 7 permutations and combinations

Unit: Probability6-7: Permutations and Combinations

Essential Question: How is a combination different from a permutation?

6 7 combinations and permutations
6-7: Combinations and Permutations
  • Permutation: an arrangement of items in a particular order
  • When all items of a particular set are used, you can alternatively use factorial notation.
    • Example:
      • In how many different ways can ten dogs line up to be groomed?
    • Answer:
      • There are 10 dogs to choose from first, multiplied by 9 dogs remaining to choose second, followed by 8 dogs to choose from to go third, etc.
      • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 10! = 3,628,800 ways
    • Your turn: In how many ways can you line up 6 trophies on a shelf?

720 ways

6 7 combinations and permutations1
6-7: Combinations and Permutations
  • Sometimes, not all items will be used. In this case, we can use the formula for permutations.
    • Example:
      • Seven yachts enter a race. First, second, and third places will be given. How many arrangements of 1st, 2nd, and 3rd places are possible for the seven yachts?
    • Answer:
      • 7 possible 1st place finishers, 6 remaining 2nd place finishers, 5 possible 3rd place finishers, means 7 • 6 • 5 = 210 arrangements
    • Your turn: How many possible 1st, 2nd & 3rd place arrangements are possible with 10 yachts?

720

6 7 permutations and combinations
6-7: Permutations and Combinations
  • When the order doesn’t matter, we use combinations
  • The only difference (mathematically) between nCr and nPr is the addition of r! in the denominator
    • Example:
      • Evaluate
6 7 permutations and combinations1
6-7: Permutations and Combinations
  • Example #4: A reading list for a course in world literature has 20 books on it. In how many ways can you choose four books to read?
  • Answer: 20 books, choosing 4 = 20C4 = 4845
  • Your turn
    • Evaluate 10C5
    • Of 20 books, in how many ways can you choose seven books?

252

77,520 ways

6 7 permutations and combinations2
6-7: Permutations and Combinations
  • Example: Ten candidates are running for three seats in the student government. You may vote for as many as three candidates. In how many ways can you vote for three or fewer candidates?
  • Answer:
    • If you vote for three people, it’s 10C3 = 120 ways
    • If you vote for two people, it’s 10C2 = 45 ways
    • If you vote for on person, it’s 10C1 = 10 ways
    • If you vote for no one, it’s 10C0 = 1 way
    • 120 + 45 + 10 + 1 = 176 different ways
  • Your turn: In how many ways can you vote for five or fewer people?

638 ways

6 7 permutations and combinations3
6-7: Permutations and Combinations
  • Worksheet
    • Problems 1 – 27
    • Odd problems